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Tarski Grothendieck Set Theory

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Tarski–Grothendieck set theory From Wikipedia, the free encyclopedia
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Page 1: Tarski Grothendieck Set Theory

Tarski–Grothendieck set theoryFrom Wikipedia, the free encyclopedia

Page 2: Tarski Grothendieck Set Theory

Contents

1 Countable set 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Formal definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Empty product 92.1 Nullary arithmetic product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Relevance of defining empty products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Nullary Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Nullary Cartesian product of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Nullary categorical product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 In logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 In computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Empty set 133.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

i

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3.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Fuzzy set 194.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Fuzzy number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Fuzzy interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Fuzzy relation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 Axiomatic definition of credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 Credibility inversion theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.8 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.9 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.10 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.13 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Hereditarily finite set 265.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Ackermann’s bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Rado graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Infinite set 286.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Recursive set 307.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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7.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8 Subset 328.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 ⊂ and ⊃ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

9 Tarski–Grothendieck set theory 369.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.2 Implementation in the Mizar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.3 Implementation in Metamath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10 Transitive set 3910.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.3 Transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.4 Transitive models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11 Uncountable set 4111.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.4 Without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12 Universal set 4312.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12.1.1 Russell’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.1.2 Cantor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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12.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

12.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 46

12.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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Chapter 1

Countable set

“Countable” redirects here. For the linguistic concept, see Count noun.Not to be confused with (recursively) enumerable sets.

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the setof natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, theelements of a countable set can always be counted one at a time and, although the counting may never finish, everyelement of the set is associated with a natural number.Some authors use countable set to mean infinitely countable alone.[1] To avoid this ambiguity, the term at mostcountable may be used when finite sets are included and countably infinite, enumerable, or denumerable[2] oth-erwise.The term countable set was originated by Georg Cantor who contrasted sets which are countable with those which areuncountable (a.k.a. nonenumerable and nondenumerable[3]). Today, countable sets are researched by a branch ofmathematics called discrete mathematics.

1.1 Definition

A set S is called countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}.[4]

If such an f can be found which is also surjective (and therefore bijective), then S is called countably infinite.In other words, a set is called “countably infinite” if it has one-to-one correspondence with the natural number set, N.As noted above, this terminology is not universal: Some authors use countable to mean what is here called “countablyinfinite,” and to not include finite sets.For alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function, seethe section Formal definition and properties below.

1.2 History

In the western world, different infinities were first classified by Georg Cantor around 1874.[5]

1.3 Introduction

A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements;for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}. This is only effective for smallsets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element,sometimes an ellipsis ("...”) is used, if the writer believes that the reader can easily guess what is missing; for example,

1

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2 CHAPTER 1. COUNTABLE SET

{1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possibleto list all the elements, because the set is finite.Some sets are infinite; these sets have more than n elements for any integer n. For example, the set of natural numbers,denotable by {0, 1, 2, 3, 4, 5, ...}, has infinitely many elements, and we cannot use any normal number to give itssize. Nonetheless, it turns out that infinite sets do have a well-defined notion of size (or more properly, of cardinality,which is the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.

YX123

x

246

2x

. .

. .

Bijective mapping from integer to even numbers

To understand what this means, we first examine what it does not mean. For example, there are infinitely many oddintegers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that thenumber of even integers, which is the same as the number of odd integers, is also the same as the number of integersoverall. This is because we arrange things such that for every integer, there is a distinct even integer: ... −2→−4,−1→−2, 0→0, 1→2, 2→4, ...; or, more generally, n→2n, see picture. What we have done here is arranged the integersand the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two setssuch that each element of each set corresponds to a single element in the other set.However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept)demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.A set is countable if: (1) it is finite, or (2) it has the same cardinality (size) as the set of natural numbers. Equivalently, aset is countable if it has the same cardinality as some subset of the set of natural numbers. Otherwise, it is uncountable.

1.4 Formal definition and properties

By definition a set S is countable if there exists an injective function f : S → N from S to the natural numbers N ={0, 1, 2, 3, ...}.

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1.4. FORMAL DEFINITION AND PROPERTIES 3

It might seem natural to divide the sets into different classes: put all the sets containing one element together; all thesets containing two elements together; ...; finally, put together all infinite sets and consider them as having the samesize. This view is not tenable, however, under the natural definition of size.To elaborate this we need the concept of a bijection. Although a “bijection” seems a more advanced concept than anumber, the usual development of mathematics in terms of set theory defines functions before numbers, as they arebased on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

a↔ 1, b↔ 2, c↔ 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this defines abijection.We now generalize this situation and define two sets to be of the same size if (and only if) there is a bijection betweenthem. For all finite sets this gives us the usual definition of “the same size”. What does it tell us about the size ofinfinite sets?Consider the sets A = {1, 2, 3, ... }, the set of positive integers and B = {2, 4, 6, ... }, the set of even positive integers.We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recallthat to prove this we need to exhibit a bijection between them. But this is easy, using n↔ 2n, so that

1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ....

As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa.Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets,a situation which is impossible for finite sets.Likewise, the set of all ordered pairs of natural numbers is countably infinite, as can be seen by following a path likethe one in the picture:The resulting mapping is like this:

0 ↔ (0,0), 1 ↔ (1,0), 2 ↔ (0,1), 3 ↔ (2,0), 4 ↔ (1,1), 5 ↔ (0,2), 6 ↔ (3,0) ....

It is evident that this mapping will cover all such ordered pairs.Interestingly: if you treat each pair as being the numerator and denominator of a vulgar fraction, then for everypositive fraction, we can come up with a distinct number corresponding to it. This representation includes also thenatural numbers, since every natural number is also a fraction N/1. So we can conclude that there are exactly as manypositive rational numbers as there are positive integers. This is true also for all rational numbers, as can be seen below(a more complex presentation is needed to deal with negative numbers).Theorem: The Cartesian product of finitely many countable sets is countable.This form of triangular mapping recursively generalizes to vectors of finitely many natural numbers by repeatedlymapping the first two elements to a natural number. For example, (0,2,3) maps to (5,3) which maps to 39.Sometimes more than onemapping is useful. This is where youmap the set which you want to show countably infinite,onto another set; and then map this other set to the natural numbers. For example, the positive rational numbers caneasily be mapped to (a subset of) the pairs of natural numbers because p/q maps to (p, q).What about infinite subsets of countably infinite sets? Do these have fewer elements than N?Theorem: Every subset of a countable set is countable. In particular, every infinite subset of a countably infinite setis countably infinite.For example, the set of prime numbers is countable, by mapping the n-th prime number to n:

• 2 maps to 1

• 3 maps to 2

• 5 maps to 3

• 7 maps to 4

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4 CHAPTER 1. COUNTABLE SET

1

2

3

01 2 31

2

3

4

5

6

7

8

9

11

12

13 18

17

24

0

0

The Cantor pairing function assigns one natural number to each pair of natural numbers

• 11 maps to 5

• 13 maps to 6

• 17 maps to 7

• 19 maps to 8

• 23 maps to 9

• ...

What about sets being “larger than” N? An obvious place to look would be Q, the set of all rational numbers, whichintuitively may seem much bigger than N. But looks can be deceiving, for we assert:Theorem: Q (the set of all rational numbers) is countable.Q can be defined as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto thesubset of ordered triples of natural numbers (a, b, c) such that a ≥ 0, b > 0, a and b are coprime, and c ∈ {0, 1} suchthat c = 0 if a/b ≥ 0 and c = 1 otherwise.

• 0 maps to (0,1,0)

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1.4. FORMAL DEFINITION AND PROPERTIES 5

• 1 maps to (1,1,0)

• −1 maps to (1,1,1)

• 1/2 maps to (1,2,0)

• −1/2 maps to (1,2,1)

• 2 maps to (2,1,0)

• −2 maps to (2,1,1)

• 1/3 maps to (1,3,0)

• −1/3 maps to (1,3,1)

• 3 maps to (3,1,0)

• −3 maps to (3,1,1)

• 1/4 maps to (1,4,0)

• −1/4 maps to (1,4,1)

• 2/3 maps to (2,3,0)

• −2/3 maps to (2,3,1)

• 3/2 maps to (3,2,0)

• −3/2 maps to (3,2,1)

• 4 maps to (4,1,0)

• −4 maps to (4,1,1)

• ...

By a similar development, the set of algebraic numbers is countable, and so is the set of definable numbers.Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable.For example, given countable sets a, b, c ...Using a variant of the triangular enumeration we saw above:

• a0 maps to 0

• a1 maps to 1

• b0 maps to 2

• a2 maps to 3

• b1 maps to 4

• c0 maps to 5

• a3 maps to 6

• b2 maps to 7

• c1 maps to 8

• d0 maps to 9

• a4 maps to 10

• ...

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6 CHAPTER 1. COUNTABLE SET

Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore alsocountable by a previous theorem.Also note that the axiom of countable choice is needed in order to index all of the sets a, b, c,...Theorem: The set of all finite-length sequences of natural numbers is countable.This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which isa countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which iscountable by the previous theorem.Theorem: The set of all finite subsets of the natural numbers is countable.If you have a finite subset, you can order the elements into a finite sequence. There are only countably many finitesequences, so also there are only countably many finite subsets.The following theorem gives equivalent formulations in terms of a bijective function or a surjective function. A proofof this result can be found in Lang’s text.[2]

Theorem: Let S be a set. The following statements are equivalent:

1. S is countable, i.e. there exists an injective function f : S → N.

2. Either S is empty or there exists a surjective function g : N→ S.

3. Either S is finite or there exists a bijection h : N→ S.

Several standard properties follow easily from this theorem. We present them here tersely. For a gentler presentationsee the sections above. Observe that N in the theorem can be replaced with any countably infinite set. In particularwe have the following Corollary.Corollary: Let S and T be sets.

1. If the function f : S → T is injective and T is countable then S is countable.

2. If the function g : S → T is surjective and S is countable then T is countable.

Proof: For (1) observe that if T is countable there is an injective function h : T → N. Then if f : S → T is injectivethe composition h o f : S → N is injective, so S is countable.For (2) observe that if S is countable there is a surjective function h : N → S. Then if g : S → T is surjective thecomposition g o h : N→ T is surjective, so T is countable.Proposition: Any subset of a countable set is countable.Proof: The restriction of an injective function to a subset of its domain is still injective.Proposition: The Cartesian product of two countable sets A and B is countable.Proof: Note that N × N is countable as a consequence of the definition because the function f : N × N → N givenby f(m, n) = 2m3n is injective. It then follows from the Basic Theorem and the Corollary that the Cartesian productof any two countable sets is countable. This follows because if A and B are countable there are surjections f : N→A and g : N→ B. So

f × g : N × N→ A × B

is a surjection from the countable set N × N to the set A × B and the Corollary implies A × B is countable. This resultgeneralizes to the Cartesian product of any finite collection of countable sets and the proof follows by induction onthe number of sets in the collection.Proposition: The integers Z are countable and the rational numbers Q are countable.Proof: The integers Z are countable because the function f : Z→ N given by f(n) = 2n if n is non-negative and f(n)= 3|n| if n is negative is an injective function. The rational numbers Q are countable because the function g : Z × N→ Q given by g(m, n) = m/(n + 1) is a surjection from the countable set Z × N to the rationals Q.Proposition: If An is a countable set for each n in N then the union of all An is also countable.

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1.5. MINIMAL MODEL OF SET THEORY IS COUNTABLE 7

Proof: This is a consequence of the fact that for each n there is a surjective function gn : N → An and hence thefunction

G : N× N →∪n∈N

An

given by G(n, m) = gn(m) is a surjection. Since N × N is countable, the Corollary implies that the union is countable.We are using the axiom of countable choice in this proof in order to pick for each n in N a surjection gn from thenon-empty collection of surjections from N to An.Cantor’s Theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is nosurjective function from A to P(A). A proof is given in the article Cantor’s Theorem. As an immediate consequenceof this and the Basic Theorem above we have:Proposition: The set P(N) is not countable; i.e. it is uncountable.For an elaboration of this result see Cantor’s diagonal argument.The set of real numbers is uncountable (see Cantor’s first uncountability proof), and so is the set of all infinitesequences of natural numbers. A topological proof for the uncountability of the real numbers is described at finiteintersection property.

1.5 Minimal model of set theory is countable

If there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standardmodel (see Constructible universe). The Löwenheim-Skolem theorem can be used to show that this minimal modelis countable. The fact that the notion of “uncountability” makes sense even in this model, and in particular that thismodel M contains elements which are

• subsets of M, hence countable,

• but uncountable from the point of view of M,

was seen as paradoxical in the early days of set theory, see Skolem’s paradox.The minimal standard model includes all the algebraic numbers and all effectively computable transcendental num-bers, as well as many other kinds of numbers.

1.6 Total orders

Countable sets can be totally ordered in various ways, e.g.:

• Well orders (see also ordinal number):

• The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)• The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)

• Other (not well orders):

• The usual order of integers (..., −3, −2, −1, 0, 1, 2, 3, ...)• The usual order of rational numbers (Cannot be explicitly written as a list!)

Note that in both examples of well orders here, any subset has a least element; and in both examples of non-wellorders, some subsets do not have a least element. This is the key definition that determines whether a total order isalso a well order.

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1.7 See also• Aleph number

• Counting

• Hilbert’s paradox of the Grand Hotel

• Uncountable set

1.8 Notes[1] For an example of this usage see (Rudin 1976, Chapter 2).

[2] See (Lang 1993, §2 of Chapter I).

[3] See (Apostol 1969, Chapter 13.19).

[4] Since there is an obvious bijection between N and N* = {1, 2, 3, ...}, it makes no difference whether one considers 0 tobe a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic,which make 0 a natural number.

[5] Stillwell, John C. (2010), Roads to Infinity: TheMathematics of Truth and Proof, CRC Press, p. 10, ISBN 9781439865507,Cantor’s discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before1874, infinity was not even considered a legitimate mathematical subject by most people, so the need to distinguish betweencountable and uncountable infinities could not have been imagined.

1.9 References• Lang, Serge (1993), Real and Functional Analysis, Berlin, New York: Springer-Verlag, ISBN 0-387-94001-4

• Rudin, Walter (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, ISBN 0-07-054235-X

• Apostol, Tom M. (June 1969),Multi-Variable Calculus and Linear Algebra with Applications, Calculus 2 (2nded.), New York: John Wiley + Sons, ISBN 978-0-471-00007-5

1.10 External links• Weisstein, Eric W., “Countable Set”, MathWorld.

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Chapter 2

Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by conventionequal to the multiplicative identity 1 (assuming there is an identity for the multiplication operation in question), justas the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.[1][2][3]

The term “empty product” is most often used in the above sense when discussing arithmetic operations. However,the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products incomputer programming; these are discussed below.

2.1 Nullary arithmetic product

2.1.1 Justification

Let a1, a2, a3,... be a sequence of numbers, and let

Pm =

m∏i=1

ai = a1 · · · am

be the product of the first m elements of the sequence. Then

Pm = am · Pm−1

for all m = 1,2,... provided that we use the following conventions: P1 = a1 and P0 = 1 . In other words, a “product”P1 with only one factor evaluates to that factor, while a “product” P0 with no factors at all evaluates to 1. Allowing a“product” with only one or zero factors reduces the number of cases to be considered in many mathematical formulas.Such “products” are natural starting points in induction proofs, as well as in algorithms. For these reasons, the “emptyproduct is one convention” is common practice in mathematics and computer programming.

2.1.2 Relevance of defining empty products

The notion of an empty product is useful for the same reason that the number zero and the empty set are useful: whilethey seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentationof many subjects.For example, the empty products 0! = 1 and x0 = 1 shorten Taylor series notation (see zero to the power of zero for adiscussion when x=0). Likewise, if M is an n × n matrix then M0 is the n × n identity matrix, reflecting the fact thatapplying a linear map zero times has the same effect as applying the identity map.As another example, the fundamental theorem of arithmetic says that every positive integer can be written uniquelyas a product of primes. However, if we do not allow products with only 0 or 1 factors, then the theorem (and itsproof!) become longer.[4][5]

9

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10 CHAPTER 2. EMPTY PRODUCT

More examples of the use of the empty product in mathematics may be found in the binomial theorem (which assumesand implies that x0=1 for all x), Stirling number, König’s theorem, binomial type, binomial series, difference operatorand Pochhammer symbol.

2.1.3 Logarithms

Since logarithms turn products into sums, they should map an empty product to an empty sum. So if we define theempty product to be 1, then the empty sum should be ln(1) = 0 . Conversely, the exponential function turns sumsinto products, so if we define the empty sum to be 0, then the empty product should be e0 = 1 .

∏i

xi = e∑

i ln xi

2.2 Nullary Cartesian product

Consider the general definition of the Cartesian product:

∏i∈I

Xi = {g : I →∪i∈I

Xi | ∀i g(i) ∈ Xi}.

If I is empty, the only such g is the empty function f∅ , which is the unique subset of∅×∅ that is a function∅ → ∅, namely the empty subset ∅ (the only subset that ∅×∅ = ∅ has):

∏∅

= {f∅ : ∅ → ∅} = {∅}.

Thus, the cardinality of the Cartesian product of no sets is 1.Under the perhaps more familiar n-tuple interpretation,

∏∅

= {()},

that is, the singleton set containing the empty tuple. Note that in both representations the empty product has cardinality1.

2.2.1 Nullary Cartesian product of functions

The empty Cartesian product of functions is again the empty function.

2.3 Nullary categorical product

In any category, the product of an empty family is a terminal object of that category. This can be demonstrated byusing the limit definition of the product. An n-fold categorical product can be defined as the limit with respect to adiagram given by the discrete category with n objects. An empty product is then given by the limit with respect tothe empty category, which is the terminal object of the category if it exists. This definition specializes to give resultsas above. For example, in the category of sets the categorical product is the usual Cartesian product, and the terminalobject is a singleton set. In the category of groups the categorical product is the Cartesian product of groups, and theterminal object is a trivial group with one element. To obtain the usual arithmetic definition of the empty product wemust take the decategorification of the empty product in the category of finite sets.Dually, the coproduct of an empty family is an initial object. Nullary categorical products or coproducts may notexist in a given category; e.g. in the category of fields, neither exists.

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2.4. IN LOGIC 11

2.4 In logic

Classical logic defines the operation of conjunction, which is generalized to universal quantification in and predicatecalculus, and is widely known as logical multiplication because we intuitively identify true with 1 and false with 0 andour conjunction behaves as ordinary multiplier. Multipliers can have arbitrary number of inputs. In case of 0 inputs,we have empty conjunction, which is identically equal to true.This is related to another concept in logic, vacuous truth, which tells us that empty set of objects can have anyproperty. It can be explained the way that the conjunction (as part of logic in general) deals with values less or equal1. This means that longer is the conjunction, the higher is probability to end up with 0. Conjunction merely checksthe propositions and returns 0 (or false) as soon as one of propositions evaluates to false. Reducing the number ofconjoined propositions increases the chance to pass the check and stay with 1. Particularly, if there are 0 tests ormembers to check, none can fail so, by default, we must always succeed regardless of which propositions or memberproperties had to be tested.

2.5 In computer programming

Many programming languages, such as Python, allow the direct expression of lists of numbers, and even functionsthat allow an arbitrary number of parameters. If such a language has a function that returns the product of all thenumbers in a list, it usually works like this:listprod( [2,3,5] ) --> 30 listprod( [2,3] ) --> 6 listprod( [2] ) --> 2 listprod( [] ) --> 1This convention helps avoid having to code special cases like “if length of list is 1” or “if length of list is zero” asspecial cases.Multiplication is an infix operator and therefore a binary operator, complicating the notation of an empty product.Some programming languages handle this by implementing variadic functions. For example, the fully parenthesizedprefix notation of Lisp languages gives rise to a natural notation for nullary functions:(* 2 2 2) ; evaluates to 8 (* 2 2) ; evaluates to 4 (* 2) ; evaluates to 2 (*) ; evaluates to 1

2.6 See also

• Iterated binary operation

• Empty sum

2.7 References

[1] Jaroslav Nešetřil, Jiří Matoušek (1998). Invitation to Discrete Mathematics. Oxford University Press. p. 12. ISBN 0-19-850207-9.

[2] A.E. Ingham and R C Vaughan (1990). The Distribution of Prime Numbers. Cambridge University Press. p. 1. ISBN0-521-39789-8.

[3] Page 9 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

[4] EdsgerWybe Dijkstra (1990-03-04). “How Computing Science created a newmathematical style”. EWD. Retrieved 2010-01-20. Hardy and Wright: “Every positive integer, except 1, is a product of primes”, Harold M. Stark: “If n is an integergreater than 1, then either n is prime or n is a finite product of primes.”. These examples —which I owe to A.J.M. vanGasteren— both reject the empty product, the last one also rejects the product with a single factor.

[5] Edsger Wybe Dijkstra (1986-11-14). “The nature of my research and why I do it”. EWD. Retrieved 2010-07-03. Butalso 0 is certainly finite and by defining the product of 0 factors —how else?— to be equal to 1 we can do away with theexception: “If n is a positive integer, then n is a finite product of primes.”

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12 CHAPTER 2. EMPTY PRODUCT

2.8 External links• PlanetMath article on the empty product

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Chapter 3

Empty set

"∅" redirects here. For similar symbols, see Ø (disambiguation).In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size orcardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists byincluding an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of setsare trivially true for the empty set.Null set was once a common synonym for “empty set”, but is now a technical term in measure theory. The empty setmay also be called the void set.

3.1 Notation

Common notations for the empty set include "{}", "∅", and " ∅ ". The latter two symbols were introduced by theBourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets(and not related in any way to the Greek letter Φ).[1]

The empty-set symbol ∅ is found at Unicode point U+2205.[2] In TeX, it is coded as \emptyset or \varnothing.

3.2 Properties

In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements;therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of “the emptyset” rather than “an empty set”.The mathematical symbols employed below are explained here.For any set A:

• The empty set is a subset of A:

∀A : ∅ ⊆ A

• The union of A with the empty set is A:

∀A : A ∪ ∅ = A

• The intersection of A with the empty set is the empty set:

∀A : A ∩ ∅ = ∅

• The Cartesian product of A and the empty set is the empty set:

∀A : A× ∅ = ∅

13

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14 CHAPTER 3. EMPTY SET

The empty set is the set containing no elements.

The empty set has the following properties:

• Its only subset is the empty set itself:

∀A : A ⊆ ∅ ⇒ A = ∅

• The power set of the empty set is the set containing only the empty set:

2∅ = {∅}

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3.2. PROPERTIES 15

A symbol for the empty set

• Its number of elements (that is, its cardinality) is zero:

card(∅) = 0

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition ofnatural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.For any property:

• For every element of ∅ the property holds (vacuous truth);

• There is no element of ∅ for which the property holds.

Conversely, if for some property and some set V, the following two statements hold:

• For every element of V the property holds;

• There is no element of V for which the property holds,

V = ∅

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16 CHAPTER 3. EMPTY SET

By the definition of subset, the empty set is a subset of any set A, as every element x of ∅ belongs to A. If it is nottrue that every element of ∅ is in A, there must be at least one element of ∅ that is not present in A. Since there areno elements of ∅ at all, there is no element of ∅ that is not in A. Hence every element of ∅ is in A, and ∅ is a subsetof A. Any statement that begins “for every element of ∅ " is not making any substantive claim; it is a vacuous truth.This is often paraphrased as “everything is true of the elements of the empty set.”

3.2.1 Operations on the empty set

Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sumof the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product).Ultimately, the results of these operations say more about the operation in question than about the empty set. Forinstance, zero is the identity element for addition, and one is the identity element for multiplication.A disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is adisarrangment of itself as no element can be found that retains its original position.

3.3 In other areas of mathematics

3.3.1 Extended real numbers

Since the empty set has no members, when it is considered as a subset of any ordered set, then every member ofthat set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of thereal numbers, with its usual ordering, represented by the real number line, every real number is both an upper andlower bound for the empty set.[3] When considered as a subset of the extended reals formed by adding two “numbers”or “points” to the real numbers, namely negative infinity, denoted −∞, which is defined to be less than every otherextended real number, and positive infinity, denoted +∞, which is defined to be greater than every other extendedreal number, then:

sup ∅ = min({−∞,+∞} ∪ R) = −∞,

and

inf ∅ = max({−∞,+∞} ∪ R) = +∞.

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound(inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinityis the identity element for the maximum and supremum operators, while positive infinity is the identity element forminimum and infimum.

3.3.2 Topology

Considered as a subset of the real number line (or more generally any topological space), the empty set is both closedand open; it is an example of a “clopen” set. All its boundary points (of which there are none) are in the empty set,and the set is therefore closed; while for every one of its points (of which there are again none), there is an openneighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the factthat every finite set is compact.The closure of the empty set is empty. This is known as “preservation of nullary unions.”

3.3.3 Category theory

If A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set isthe unique initial object of the category of sets and functions.

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3.4. QUESTIONED EXISTENCE 17

The empty set can be turned into a topological space, called the empty space, in just one way: by defining the emptyset to be open. This empty topological space is the unique initial object in the category of topological spaces withcontinuous maps.The empty set is more ever a strict initial object: only the empty set has a function to the empty set.

3.4 Questioned existence

3.4.1 Axiomatic set theory

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness followsfrom the axiom of extensionality. However, the axiom of empty set can be shown redundant in either of two ways:

• There is already an axiom implying the existence of at least one set. Given such an axiom together with theaxiom of separation, the existence of the empty set is easily proved.

• In the presence of urelements, it is easy to prove that at least one set exists, viz. the set of all urelements. Again,given the axiom of separation, the empty set is easily proved.

3.4.2 Philosophical issues

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity,whose meaning and usefulness are debated by philosophers and logicians.The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something.This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explainsthat the empty set is not nothing, but rather “the set of all triangles with four sides, the set of all numbers that arebigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king.”[4]

The popular syllogism

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sand-wich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darlingwrites that the contrast can be seen by rewriting the statements “Nothing is better than eternal happiness” and "[A]ham sandwich is better than nothing” in a mathematical tone. According to Darling, the former is equivalent to “Theset of all things that are better than eternal happiness is ∅ " and the latter to “The set {ham sandwich} is better thanthe set ∅ ". It is noted that the first compares elements of sets, while the second compares the sets themselves.[4]

Jonathan Lowe argues that while the empty set:

"...was undoubtedly an important landmark in the history of mathematics, … we should not assume thatits utility in calculation is dependent upon its actually denoting some object.”

it is also the case that:

“All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and(3) is unique amongst sets in having no members. However, there are very many things that 'have nomembers’, in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things haveno members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a setwhich has no members. We cannot conjure such an entity into existence by mere stipulation.”[5]

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtainedby plural quantification over individuals, without reifying sets as singular entities having other entities as members.[6]

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18 CHAPTER 3. EMPTY SET

3.5 See also• Inhabited set

• Nothing

3.6 Notes[1] Earliest Uses of Symbols of Set Theory and Logic.

[2] Unicode Standard 5.2

[3] Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9.

[4] D. J. Darling (2004). The universal book of mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.

[5] E. J. Lowe (2005). Locke. Routledge. p. 87.

[6] • George Boolos, 1984, “To be is to be the value of a variable,” The Journal of Philosophy 91: 430–49. Reprinted inhis 1998 Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard Univ. Press: 54–72.

3.7 References• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

• Graham, Malcolm (1975), Modern Elementary Mathematics (HARDCOVER) (in English) (2nd ed.), NewYork: Harcourt Brace Jovanovich, ISBN 0155610392

3.8 External links• Weisstein, Eric W., “Empty Set”, MathWorld.

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Chapter 4

Fuzzy set

In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced byLotfi A. Zadeh[1] and Dieter Klaua[2] in 1965 as an extension of the classical notion of set. At the same time, Salii(1965) defined a more general kind of structures called L-relations, which he studied in an abstract algebraic context.Fuzzy relations, which are used now in different areas, such as linguistics (De Cock, et al., 2000), decision-making(Kuzmin, 1982) and clustering (Bezdek, 1978), are special cases of L-relations when L is the unit interval [0, 1].In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition— an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessmentof the membership of elements in a set; this is described with the aid of a membership function valued in the real unitinterval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases ofthe membership functions of fuzzy sets, if the latter only take values 0 or 1.[3] In fuzzy set theory, classical bivalentsets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which informationis incomplete or imprecise, such as bioinformatics.[4]

It has been suggested by Thayer Watkins that Zadeh’s ethnicity is an example of a fuzzy set because “His father wasTurkish-Iranian (Azerbaijani) and his mother was Russian. His father was a journalist working in Baku, Azerbaijanin the Soviet Union...Lotfi was born in Baku in 1921 and lived there until his family moved to Tehran in 1931.”[5]

4.1 Definition

A fuzzy set is a pair (U,m) where U is a set andm : U → [0, 1].

For each x ∈ U, the valuem(x) is called the grade of membership of x in (U,m). For a finite setU = {x1, . . . , xn},the fuzzy set (U,m) is often denoted by {m(x1)/x1, . . . ,m(xn)/xn}.Let x ∈ U. Then x is called not included in the fuzzy set (U,m) if m(x) = 0 , x is called fully included ifm(x) = 1 , and x is called a fuzzy member if 0 < m(x) < 1 .[6] The set {x ∈ U | m(x) > 0} is called thesupport of (U,m) and the set {x ∈ U | m(x) = 1} is called its kernel or core. The function m is called themembership function of the fuzzy set (U,m).

Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a(fixed or variable) algebra or structureL of a given kind; usually it is required thatL be at least a poset or lattice. Theseare usually called L-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membershipfunctions with values in [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizationswere first considered in 1967 by Joseph Goguen, who was a student of Zadeh.[7]

4.2 Fuzzy logic

Main article: Fuzzy logic

As an extension of the case of multi-valued logic, valuations ( µ : Vo → W ) of propositional variables ( Vo ) into aset of membership degrees (W ) can be thought of as membership functions mapping predicates into fuzzy sets (or

19

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20 CHAPTER 4. FUZZY SET

more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logiccan be extended to allow for fuzzy premises from which graded conclusions may be drawn.[8]

This extension is sometimes called “fuzzy logic in the narrow sense” as opposed to “fuzzy logic in the wider sense,”which originated in the engineering fields of automated control and knowledge engineering, and which encompassesmany topics involving fuzzy sets and “approximated reasoning.”[9]

Industrial applications of fuzzy sets in the context of “fuzzy logic in the wider sense” can be found at fuzzy logic.

4.3 Fuzzy number

Main article: Fuzzy number

A fuzzy number is a convex, normalized fuzzy set A ⊆ R whose membership function is at least segmentallycontinuous and has the functional value µA(x) = 1 at precisely one element.This can be likened to the funfair game “guess your weight,” where someone guesses the contestant’s weight, withcloser guesses being more correct, and where the guesser “wins” if he or she guesses near enough to the contestant’sweight, with the actual weight being completely correct (mapping to 1 by the membership function).

4.4 Fuzzy interval

A fuzzy interval is an uncertain set A ⊆ R with a mean interval whose elements possess the membership functionvalue µA(x) = 1 . As in fuzzy numbers, the membership function must be convex, normalized, at least segmentallycontinuous.[10]

4.5 Fuzzy relation equation

The fuzzy relation equation is an equation of the form A · R = B, where A and B are fuzzy sets, R is a fuzzy relation,and A · R stands for the composition of A with R.

4.6 Axiomatic definition of credibility[11] Let A be a non-empty set and P(A) be the power set of A . The set function Cr is known as credibility measureif it satisfies following condition

• Axiom 1: Cr{A} = 1

• Axiom 2: If B is subset of C, then, Cr{B} ≤ Cr{C}

• Axiom 3: Cr{B}+ Cr{Bc} = 1

• Axiom 4: Cr{∪Ai} = supi(Cr(Ai)) , for any event Ai with supi Cr{Ai} < 0.5

Cr{B} indicates how frequently event B will occur.

4.7 Credibility inversion theorem[12] Let A be a fuzzy variable with membership function u. Then for any set B of real numbers, we have

Cr{A ∈ B} =1

2

(supt∈B

u(t) + 1− supt∈Bc

u(t)

)

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4.8. EXPECTED VALUE 21

4.8 Expected Value[13] Let A be a fuzzy variable. Then the expected value is

E[A] =

∫ ∞

0

Cr{A ≥ t} dt−∫ 0

−∞Cr{A ≤ t} dt.

4.9 Entropy[14] Let A be a fuzzy variable with a continuous membership function. Then its entropy is

H[A] =

∫ ∞

−∞S(Cr{A ≥ t}) dt.

Where

S(y) = −y lny − (1− y) ln(1− y)

4.10 Generalizations

There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were intro-duced in 1965, a lot of new mathematical constructions and theories treating imprecision, inexactness, ambiguity,and uncertainty have been developed. Some of these constructions and theories are extensions of fuzzy set theory,while others try to mathematically model imprecision and uncertainty in a different way (Burgin and Chunihin, 1997;Kerre, 2001; Deschrijver and Kerre, 2003).The diversity of such constructions and corresponding theories includes:

• interval sets (Moore, 1966),

• L-fuzzy sets (Goguen, 1967),

• flou sets (Gentilhomme, 1968),

• Boolean-valued fuzzy sets (Brown, 1971),

• type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975),

• set-valued sets (Chapin, 1974; 1975),

• interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),

• functions as generalizations of fuzzy sets and multisets (Lake, 1976),

• level fuzzy sets (Radecki, 1977)

• underdetermined sets (Narinyani, 1980),

• rough sets (Pawlak, 1982),

• intuitionistic fuzzy sets (Atanassov, 1983),

• fuzzy multisets (Yager, 1986),

• intuitionistic L-fuzzy sets (Atanassov, 1986),

• rough multisets (Grzymala-Busse, 1987),

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22 CHAPTER 4. FUZZY SET

• fuzzy rough sets (Nakamura, 1988),

• real-valued fuzzy sets (Blizard, 1989),

• vague sets (Wen-Lung Gau and Buehrer, 1993),

• Q-sets (Gylys, 1994)

• shadowed sets (Pedrycz, 1998),

• α-level sets (Yao, 1997),

• genuine sets (Demirci, 1999),

• soft sets (Molodtsov, 1999),

• intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003)

• blurry sets (Smith, 2004)

• L-fuzzy rough sets (Radzikowska and Kerre, 2004),

• generalized rough fuzzy sets (Feng, 2010)

• rough intuitionistic fuzzy sets (Thomas and Nair, 2011),

• soft rough fuzzy sets (Meng, Zhang and Qin, 2011)

• soft fuzzy rough sets (Meng, Zhang and Qin, 2011)

• soft multisets (Alkhazaleh, Salleh and Hassan, 2011)

• fuzzy soft multisets (Alkhazaleh and Salleh, 2012)

4.11 See also• Alternative set theory

• Defuzzification

• Fuzzy concept

• Fuzzy mathematics

• Fuzzy measure theory

• Fuzzy set operations

• Fuzzy subalgebra

• Linear partial information

• Neuro-fuzzy

• Rough fuzzy hybridization

• Rough set

• Sørensen similarity index

• Type-2 Fuzzy Sets and Systems

• Uncertainty

• Interval finite element

• Multiset

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4.12. REFERENCES 23

4.12 References[1] L. A. Zadeh (1965) “Fuzzy sets”. Information and Control 8 (3) 338–353.

[2] Klaua, D. (1965) Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876.A recent in-depth analysis of this paper has been provided by Gottwald, S. (2010). “An early approach toward graded iden-tity and graded membership in set theory”. Fuzzy Sets and Systems 161 (18): 2369–2379. doi:10.1016/j.fss.2009.12.005.

[3] D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.

[4] Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, “FM-test: AFuzzy-Set-Theory-Based Approach to Differential Gene Expression Data Analysis”, BMC Bioinformatics, 7 (Suppl 4):S7. 2006.

[5] “Fuzzy Logic: The Logic of Fuzzy Sets”

[6] AAAI

[7] Goguen, Joseph A., 196, "L-fuzzy sets”. Journal of Mathematical Analysis and Applications 18: 145–174

[8] Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies PressLtd., ISBN 978-0-86380-262-1

[9] “The concept of a linguistic variable and its application to approximate reasoning,” Information Sciences 8: 199–249,301–357; 9: 43–80.

[10] “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems 1: 3–28

[11] Liu, Baoding. “Uncertain theory: an introduction to its axiomatic foundations.” Berlin: Springer-Verlag (2004).

[12] Liu, Baoding, and Yian-Kui Liu. “Expected value of fuzzy variable and fuzzy expected value models.” Fuzzy Systems,IEEE Transactions on 10.4 (2002): 445-450.

[13] Liu, Baoding, and Yian-Kui Liu. “Expected value of fuzzy variable and fuzzy expected value models.” Fuzzy Systems,IEEE Transactions on 10.4 (2002): 445-450.

[14] Xuecheng, Liu. “Entropy, distancemeasure and similarity measure of fuzzy sets and their relations.” Fuzzy sets and systems52.3 (1992): 305-318.

4.13 Further reading

• Alkhazaleh, S. and Salleh, A.R. Fuzzy Soft Multiset Theory, Abstract and Applied Analysis, 2012, article ID350600, 20 p.

• Alkhazaleh, S., Salleh, A.R. and Hassan, N. Soft Multisets Theory, Applied Mathematical Sciences, v. 5, No.72, 2011, pp. 3561–3573

• Atanassov, K. T. (1983) Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposited in Central Sci.-TechnicalLibrary of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)

• Atanasov, K. (1986) Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, v. 20, No. 1, pp. 87–96

• Bezdek, J.C. (1978) Fuzzy partitions and relations and axiomatic basis for clustering, Fuzzy Sets and Systems,v.1, pp. 111–127

• Blizard, W.D. (1989) Real-valued Multisets and Fuzzy Sets, Fuzzy Sets and Systems, v. 33, pp. 77–97

• Brown, J.G. (1971) A Note on Fuzzy Sets, Information and Control, v. 18, pp. 32–39

• Chapin, E.W. (1974) Set-valued Set Theory, I, Notre Dame J. Formal Logic, v. 15, pp. 619–634

• Chapin, E.W. (1975) Set-valued Set Theory, II, Notre Dame J. Formal Logic, v. 16, pp. 255–267

• Chris Cornelis, Martine De Cock and Etienne E. Kerre, Intuitionistic fuzzy rough sets: at the crossroads ofimperfect knowledge, Expert Systems, v. 20, issue 5, pp. 260–270, 2003

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24 CHAPTER 4. FUZZY SET

• Cornelis, C., Deschrijver, C., and Kerre, E. E. (2004) Implication in intuitionistic and interval-valued fuzzy settheory: construction, classification, application, International Journal of Approximate Reasoning, v. 35, pp.55–95

• Martine De Cock, Ulrich Bodenhofer, and Etienne E. Kerre, Modelling Linguistic Expressions Using FuzzyRelations, (2000) Proceedings 6th International Conference on Soft Computing. Iizuka 2000, Iizuka, Japan(1–4 October 2000) CDROM. p. 353-360

• Demirci, M. (1999) Genuine Sets, Fuzzy Sets and Systems, v. 105, pp. 377–384

• Deschrijver, G. and Kerre, E.E. On the relationship between some extensions of fuzzy set theory, Fuzzy Setsand Systems, v. 133, no. 2, pp. 227–235, 2003

• Didier Dubois, Henri M. Prade, ed. (2000). Fundamentals of fuzzy sets. The Handbooks of Fuzzy Sets Series7. Springer. ISBN 978-0-7923-7732-0.

• Feng F. Generalized Rough Fuzzy Sets Based on Soft Sets, Soft Computing, July 2010, Volume 14, Issue 9,pp 899–911

• Gentilhomme, Y. (1968) Les ensembles flous en linguistique, Cahiers Linguistique Theoretique Appliqee, 5,pp. 47–63

• Gogen, J.A. (1967) L-fuzzy Sets, Journal Math. Analysis Appl., v. 18, pp. 145–174

• Gottwald, S. (2006). “Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Basedand Axiomatic Approaches”. Studia Logica 82 (2): 211–244. doi:10.1007/s11225-006-7197-8.. Gottwald,S. (2006). “Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category TheoreticApproaches”. Studia Logica 84: 23–50. doi:10.1007/s11225-006-9001-1. preprint..

• Grattan-Guinness, I. (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z. Math.Logik. Grundladen Math. 22, pp. 149–160.

• Grzymala-Busse, J. Learning from examples based on rough multisets, in Proceedings of the 2nd InternationalSymposium on Methodologies for Intelligent Systems, Charlotte, NC, USA, 1987, pp. 325–332

• Gylys, R. P. (1994) Quantal sets and sheaves over quantales, Liet. Matem. Rink., v. 34, No. 1, pp. 9–31.

• Ulrich Höhle, Stephen Ernest Rodabaugh, ed. (1999). Mathematics of fuzzy sets: logic, topology, and measuretheory. The Handbooks of Fuzzy Sets Series 3. Springer. ISBN 978-0-7923-8388-8.

• Jahn, K. U. (1975) Intervall-wertige Mengen, Math.Nach. 68, pp. 115–132

• Kerre, E.E. A first view on the alternatives of fuzzy set theory, Computational Intelligence in Theory andPractice (B. Reusch, K-H . Temme, eds) Physica-Verlag, Heidelberg (ISBN 3-7908-1357-5), 2001, pp. 55–72

• George J. Klir; Bo Yuan (1995). Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall. ISBN978-0-13-101171-7.

• Kuzmin,V.B. Building Group Decisions in Spaces of Strict and Fuzzy Binary Relations, Nauka, Moscow, 1982(in Russian)

• Lake, J. (1976) Sets, fuzzy sets, multisets and functions, J. London Math. Soc., II Ser., v. 12, pp. 323–326

• Meng, D., Zhang, X. and Qin, K. Soft rough fuzzy sets and soft fuzzy rough sets, 'Computers & Mathematicswith Applications’, v. 62, issue 12, 2011, pp. 4635–4645

• Miyamoto, S. Fuzzy Multisets and their Generalizations, in 'Multiset Processing', LNCS 2235, pp. 225–235,2001

• Molodtsov, O. (1999) Soft set theory – first results, Computers & Mathematics with Applications, v. 37, No.4/5, pp. 19–31

• Moore, R.E. Interval Analysis, New York, Prentice-Hall, 1966

• Nakamura, A. (1988) Fuzzy rough sets, 'Notes on Multiple-valued Logic in Japan', v. 9, pp. 1–8

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4.14. EXTERNAL LINKS 25

• Narinyani, A.S. Underdetermined Sets – A new datatype for knowledge representation, Preprint 232, ProjectVOSTOK, issue 4, Novosibirsk, Computing Center, USSR Academy of Sciences, 1980

• Pedrycz, W. Shadowed sets: representing and processing fuzzy sets, IEEE Transactions on System, Man, andCybernetics, Part B, 28, 103-109, 1998.

• Radecki, T. Level Fuzzy Sets, 'Journal of Cybernetics’, Volume 7, Issue 3-4, 1977

• Radzikowska, A.M. and Etienne E. Kerre, E.E. On L-Fuzzy Rough Sets, Artificial Intelligence and SoftComputing - ICAISC 2004, 7th International Conference, Zakopane, Poland, June 7–11, 2004, Proceedings;01/2004

• Salii, V.N. (1965) Binary L-relations, Izv. Vysh. Uchebn. Zaved., Matematika, v. 44, No.1, pp. 133–145 (inRussian)

• Sambuc, R. Fonctions φ-floues: Application a l'aide au diagnostic en pathologie thyroidienne, Ph. D. ThesisUniv. Marseille, France, 1975.

• Seising, Rudolf: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications—Developments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, [et al.]:Springer 2007.

• Smith, N.J.J. (2004) Vagueness and blurry sets, 'J. of Phil. Logic', 33, pp. 165–235

• Thomas, K.V. and L. S. Nair, Rough intuitionistic fuzzy sets in a lattice, 'International Mathematical Forum',Vol. 6, 2011, no. 27, 1327 - 1335

• Yager, R. R. (1986) On the Theory of Bags, International Journal of General Systems, v. 13, pp. 23–37

• Yao, Y.Y., Combination of rough and fuzzy sets based on α-level sets, in: Rough Sets and Data Mining:Analysis for Imprecise Data, Lin, T.Y. and Cercone, N. (Eds.), Kluwer Academic Publishers, Boston, pp.301–321, 1997.

• Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, v. 109, Issue 1-4, 1998,pp. 227 – 242

• Zadeh, L. (1975) The concept of a linguistic variable and its application to approximate reasoning–I, Inform.Sci., v. 8, pp. 199–249

• Hans-Jürgen Zimmermann (2001). Fuzzy set theory—and its applications (4th ed.). Kluwer. ISBN 978-0-7923-7435-0.

• Gianpiero Cattaneo and Davide Ciucci, “Heyting Wajsberg Algebras as an Abstract Environment LinkingFuzzy andRough Sets” in J.J. Alpigini et al. (Eds.): RSCTC2002, LNAI 2475, pp. 77–84, 2002. doi:10.1007/3-540-45813-1_10

4.14 External links• Uncertainty model Fuzziness

• Fuzzy Systems Journal

• ScholarPedia

• The Algorithm of Fuzzy Analysis

• Fuzzy Image Processing

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Chapter 5

Hereditarily finite set

“Nested set” redirects here. Nested set may also refer to the Nested set model in relational databases.In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily

V4 represented with circles in place of curly brackets

finite sets.

5.1 Formal definition

A recursive definition of well-founded hereditarily finite sets goes as follows:

Base case: The empty set is a hereditarily finite set.Recursion rule: If a1,...,ak are hereditarily finite, then so is {a1,...,ak}.

The set of all well-founded hereditarily finite sets is denoted Vω. If we denote P(S) for the power set of S, Vω canalso be constructed by first taking the empty set written V0, then V1 = P(V0), V2 = P(V1),..., Vk = P(Vk₋₁),... Then

∞∪k=0

Vk = Vω.

5.2 Discussion

The hereditarily finite sets are a subclass of the Von Neumann universe. They are a model of the axioms consistingof the axioms of set theory with the axiom of infinity replaced by its negation, thus proving that the axiom of infinityis not a consequence of the other axioms of set theory.

26

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5.3. ACKERMANN’S BIJECTION 27

Notice that there are countably many hereditarily finite sets, since Vn is finite for any finite n (its cardinality is n−12,see tetration), and the union of countably many finite sets is countable.Equivalently, a set is hereditarily finite if and only if its transitive closure is finite. Vω is also symbolized by Hℵ0 ,meaning hereditarily of cardinality less than ℵ0 .

5.3 Ackermann’s bijection

Ackermann (1937) gave the following natural bijection f from the natural numbers to the hereditarily finite sets,known as the Ackermann coding. It is defined recursively by

f(2a + 2b + · · · ) = {f(a), f(b), . . .} if a, b, ... are distinct.

We have f(m)∈f(n) if and only if the mth binary digit of n (counting from the right starting at 0) is 1.

5.4 Rado graph

The graph whose vertices are the hereditarily finite sets, with an edge joining two vertices whenever one is containedin the other, is the Rado graph or random graph.

5.5 See also• Hereditarily countable set

5.6 References• Ackermann, Wilhelm (1937), “Die Widerspruchsfreiheit der allgemeinen Mengenlehre”, Mathematische An-nalen 114 (1): 305–315, doi:10.1007/BF01594179

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Chapter 6

Infinite set

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Someexamples are:

• the set of all integers, {..., −1, 0, 1, 2, ...}, is a countably infinite set; and

• the set of all real numbers is an uncountably infinite set.

6.1 Properties

The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set thatis directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers.A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number.If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset.If a set of sets is infinite or contains an infinite element, then its union is infinite. The powerset of an infinite set isinfinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then atleast one of themmust be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian productof an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets each containingat least two elements is either empty or infinite; if the axiom of choice holds, then it is infinite.If an infinite set is a well-ordered set, then it must have a nonempty subset that has no greatest element.In ZF, a set is infinite if and only if the powerset of its powerset is a Dedekind-infinite set, having a proper subsetequinumerous to itself.[1] If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.

6.2 See also

• Aleph number

• Dedekind-infinite set

• Infinity

6.3 References[1] Boolos, George (1994), “The advantages of honest toil over theft”, Mathematics and mind (Amherst, MA, 1991), Logic

Comput. Philos., Oxford Univ. Press, New York, pp. 27–44, MR 1373892. See in particular pp. 32–33.

28

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6.4. EXTERNAL LINKS 29

6.4 External links• Weisstein, Eric W., “Infinite Set”, MathWorld.

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Chapter 7

Recursive set

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithmwhich terminates after a finite amount of time and correctly decides whether or not a given number belongs to theset.A more general class of sets consists of the recursively enumerable sets, also called semidecidable sets. For thesesets, it is only required that there is an algorithm that correctly decides when a number is in the set; the algorithmmay give no answer (but not the wrong answer) for numbers not in the set.A set which is not computable is called noncomputable or undecidable.

7.1 Formal definition

A subset S of the natural numbers is called recursive if there exists a total computable function f such that f(x) = 1 ifx ∈ S and f(x) = 0 if x ∉ S . In other words, the set S is recursive if and only if the indicator function 1S is computable.

7.2 Examples

• Every finite or cofinite subset of the natural numbers is computable. This includes these special cases:

• The empty set is computable.• The entire set of natural numbers is computable.• Each natural number (as defined in standard set theory) is computable; that is, the set of natural numbersless than a given natural number is computable.

• The set of prime numbers is computable.

• A recursive language is a recursive subset of a formal language.

• The set of Gödel numbers of arithmetic proofs described in Kurt Gödel’s paper “On formally undecidablepropositions of Principia Mathematica and related systems I"; see Gödel’s incompleteness theorems.

7.3 Properties

If A is a recursive set then the complement of A is a recursive set. If A and B are recursive sets then A ∩ B, A ∪ Band the image of A × B under the Cantor pairing function are recursive sets.A setA is a recursive set if and only ifA and the complement ofA are both recursively enumerable sets. The preimageof a recursive set under a total computable function is a recursive set. The image of a computable set under a totalcomputable bijection is computable.

30

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7.4. REFERENCES 31

A set is recursive if and only if it is at level Δ01 of the arithmetical hierarchy.A set is recursive if and only if it is either the range of a nondecreasing total computable function or the empty set.The image of a computable set under a nondecreasing total computable function is computable.

7.4 References• Cutland, N. Computability. Cambridge University Press, Cambridge-New York, 1980. ISBN 0-521-22384-9;ISBN 0-521-29465-7

• Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1;ISBN 0-07-053522-1

• Soare, R. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag,Berlin, 1987. ISBN 3-540-15299-7

7.5 External links• Sakharov, Alex, “Recursive Set”, MathWorld.

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Chapter 8

Subset

“Superset” redirects here. For other uses, see Superset (disambiguation).In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is

AB

Euler diagram showingA is a proper subset of B and conversely B is a proper superset of A

32

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8.1. DEFINITIONS 33

“contained” inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of oneset being a subset of another is called inclusion or sometimes containment.The subset relation defines a partial order on sets.The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

8.1 Definitions

If A and B are sets and every element of A is also an element of B, then:

• A is a subset of (or is included in) B, denoted by A ⊆ B ,or equivalently

• B is a superset of (or includes) A, denoted by B ⊇ A.

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A),then

• A is also a proper (or strict) subset of B; this is written as A ⊊ B.

or equivalently

• B is a proper superset of A; this is written as B ⊋ A.

For any set S, the inclusion relation ⊆ is a partial order on the set P(S) of all subsets of S (the power set of S).When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.[1]

8.2 ⊂ and ⊃ symbols

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaningand instead of the symbols, ⊆ and ⊇.[2] So for example, for these authors, it is true of every set A that A ⊂ A.Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, instead of ⊊ and⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may ormay not equal y, but if x < y, then x may not equal y, and is less than y. Similarly, using the convention that ⊂ isproper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B.

8.3 Examples

• The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions A ⊆ B and A ⊊ B are true.

• The set D = {1, 2, 3} is a subset of E = {1, 2, 3}, thus D ⊆ E is true, and D ⊊ E is not true (false).

• Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)

• The empty set { }, denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any setexcept itself.

• The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}

• The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a linesegment is a proper subset of the set of points in a line. These are two examples in which both the subset andthe whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is,the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.

• The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinitebut the latter set has a larger cardinality (or power) than the former set.

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34 CHAPTER 8. SUBSET

polygonsregular

polygons

The regular polygons form a subset of the polygons

Another example in an Euler diagram:

• A is a proper subset of B

• C is a subset but no proper subset of B

8.4 Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, ⪯ ) is isomorphic to somecollection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identifiedwith the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b].For the power set P(S) of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product ofk = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumeratingS = {s1, s2, …, sk} and associating with each subset T ⊆ S (which is to say with each element of 2S) the k-tuple from{0,1}k of which the ith coordinate is 1 if and only if si is a member of T.

8.5 See also• Containment order

8.6 References[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN

978-0-07-338309-5.

[2] Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1,MR 924157

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8.7. EXTERNAL LINKS 35

C B A

A B and B C imply A C

[3] Subsets and Proper Subsets (PDF), retrieved 2012-09-07

• Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

8.7 External links• Weisstein, Eric W., “Subset”, MathWorld.

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Chapter 9

Tarski–Grothendieck set theory

Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck)is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distin-guished from other axiomatic set theories by the inclusion of Tarski’s axiom which states that for each set there is aGrothendieck universe it belongs to (see below). Tarski’s axiom implies the existence of inaccessible cardinals, pro-viding a richer ontology than that of conventional set theories such as ZFC. For example, adding this axiom supportscategory theory.The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs.

9.1 Axioms

Tarski–Grothendieck set theory starts with conventional Zermelo–Fraenkel set theory and then adds “Tarski’s axiom”.We will use the axioms, definitions, and notation of Mizar to describe it. Mizar’s basic objects and processes are fullyformal; they are described informally below. First, let us assume that:

• Given any set A , the singleton {A} exists.

• Given any two sets, their unordered and ordered pairs exist.

• Given any family of sets, its union exists.

TG includes the following axioms, which are conventional because they are also part of ZFC:

• Set axiom: Quantified variables range over sets alone; everything is a set (the same ontology as ZFC).

• Extensionality axiom: Two sets are identical if they have the same members.

• Axiom of regularity: No set is a member of itself, and circular chains of membership are impossible.

• Axiom schema of replacement: Let the domain of the function F be the set A . Then the range of F (thevalues of F (x) for all members x of A ) is also a set.

It is Tarski’s axiom that distinguishes TG from other axiomatic set theories. Tarski’s axiom also implies the axiomsof infinity, choice,[1][2] and power set.[3][4] It also implies the existence of inaccessible cardinals, thanks to which theontology of TG is much richer than that of conventional set theories such as ZFC.

• Tarski’s axiom (adapted from Tarski 1939[5]). For every set x , there exists a set y whose members include:

- x itself;- every subset of every member of y ;- the power set of every member of y ;

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9.2. IMPLEMENTATION IN THE MIZAR SYSTEM 37

- every subset of y of cardinality less than that of y .More formally:

∀x∃y[x ∈ y ∧ ∀z ∈ y(P(z) ⊆ y ∧ P(z) ∈ y) ∧ ∀z ∈ P(y)(¬z ≈ y → z ∈ y)]

where " P(x) " denotes the power class of x and " ≈ " denotes equinumerosity. What Tarski’s axiom states (in thevernacular) for each set x there is a Grothendieck universe it belongs to.

9.2 Implementation in the Mizar system

The Mizar language, underlying the implementation of TG and providing its logical syntax, is typed and the typesare assumed to be non-empty. Hence, the theory is implicitly taken to be non-empty. The existence axioms, e.g. theexistence of the unordered pair, is also implemented indirectly by the definition of term constructors.The system includes equality, the membership predicate and the following standard definitions:

• Singleton: A set with one member;

• Unordered pair: A set with two distinct members. {a, b} = {b, a} ;

• Ordered pair: The set {{a, b}, {a}} = (a, b) = (b, a) ;

• Subset: A set all of whose members are members of another given set;

• The union of a family of sets Y : The set of all members of any member of Y .

9.3 Implementation in Metamath

The Metamath system supports arbitrary higher-order logics, but it is typically used with the “set.mm” definitions ofaxioms. The ax-groth axiom adds Tarski’s axiom, which in Metamath is defined as follows:⊢ ∃y(x ∈ y ∧ ∀z ∈ y (∀w(w ⊆ z → w ∈ y) ∧ ∃w ∈ y ∀v(v ⊆ z → v ∈ w)) ∧ ∀z(z ⊆ y → (z ≈ y ∨ z ∈ y)))

9.4 See also

• Axiom of limitation of size

9.5 Notes

[1] Tarski (1938)

[2] http://mmlquery.mizar.org/mml/current/wellord2.html#T26

[3] Robert Solovay, Re: AC and strongly inaccessible cardinals.

[4] Metamath grothpw.

[5] Tarski (1939)

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38 CHAPTER 9. TARSKI–GROTHENDIECK SET THEORY

9.6 References• Andreas Blass, I.M. Dimitriou, and Benedikt Löwe (2007) "Inaccessible Cardinals without the Axiom ofChoice," Fundamenta Mathematicae 194: 179-89.

• Bourbaki, Nicolas (1972). “Univers”. In Michael Artin, Alexandre Grothendieck, Jean-Louis Verdier, eds.Séminaire de Géométrie Algébrique du Bois Marie – 1963-64 – Théorie des topos et cohomologie étale desschémas – (SGA 4) – vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag. pp. 185–217.

• Patrick Suppes (1960) Axiomatic Set Theory. Van Nostrand. Dover reprint, 1972.

• Tarski, Alfred (1938). "Über unerreichbare Kardinalzahlen” (PDF). Fundamenta Mathematicae 30: 68–89.

• Tarski, Alfred (1939). “On the well-ordered subsets of any set” (PDF). Fundamenta Mathematicae 32: 176–183.

9.7 External links• Trybulec, Andrzej, 1989, "Tarski–Grothendieck Set Theory", Journal of Formalized Mathematics.

• Metamath: "Proof Explorer Home Page." Scroll down to “Grothendieck’s Axiom.”

• PlanetMath: "Tarski’s Axiom"

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Chapter 10

Transitive set

In set theory, a set A is transitive, if and only if

• whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,• whenever x ∈ A, and x is not an urelement, then x is a subset of A.

Similarly, a class M is transitive if every element of M is a subset of M.

10.1 Examples

Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarilytransitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals).Any of the stages Vα and Lα leading to the construction of the von Neumann universe V and Gödel’s constructibleuniverse L are transitive sets. The universes L and V themselves are transitive classes.

10.2 Properties

A set X is transitive if and only if∪X ⊆ X , where

∪X is the union of all elements of X that are sets,

∪X = {y |

(∃x ∈ X)y ∈ x} . If X is transitive, then∪X is transitive. If X and Y are transitive, then X∪Y∪{X,Y} is transitive.

In general, if X is a class all of whose elements are transitive sets, then X ∪∪X is transitive.

A set X which does not contain urelements is transitive if and only if it is a subset of its own power set,X ⊂ P(X).The power set of a transitive set without urelements is transitive.

10.3 Transitive closure

The transitive closure of a set X is the smallest (with respect to inclusion) transitive set which contains X. Supposeone is given a set X, then the transitive closure of X is

∪{X,

∪X,

∪∪X,

∪∪∪X,

∪∪∪∪X, . . .}.

Note that this is the set of all of the objects related to X by the transitive closure of the membership relation.

10.4 Transitive models of set theory

Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models.The reason is that properties defined by bounded formulas are absolute for transitive classes.

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40 CHAPTER 10. TRANSITIVE SET

A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system.Transitivity is an important factor in determining the absoluteness of formulas.In the superstructure approach to non-standard analysis, the non-standard universes satisfy strong transitivity.[1]

10.5 See also• End extension

• Transitive relation

• Supertransitive class

10.6 References[1] Goldblatt (1998) p.161

• Ciesielski, Krzysztof (1997), Set theory for the working mathematician, London Mathematical Society StudentTexts 39, Cambridge: Cambridge University Press, ISBN 0-521-59441-3, Zbl 0938.03067

• Goldblatt, Robert (1998), Lectures on the hyperreals. An introduction to nonstandard analysis, Graduate Textsin Mathematics 188, New York, NY: Springer-Verlag, ISBN 0-387-98464-X, Zbl 0911.03032

• Jech, Thomas (2008) [originally published in 1973], The Axiom of Choice, Dover Publications, ISBN 0-486-46624-8, Zbl 0259.02051

10.7 External links• Weisstein, Eric W., “Transitive”, MathWorld.

• Weisstein, Eric W., “Transitive Closure”, MathWorld.

• Weisstein, Eric W., “Transitive Reduction”, MathWorld.

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Chapter 11

Uncountable set

“Uncountable” redirects here. For the linguistic concept, see Uncountable noun.

In mathematics, an uncountable set (or uncountably infinite set)[1] is an infinite set that contains too many elementsto be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinalnumber is larger than that of the set of all natural numbers.

11.1 Characterizations

There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of thefollowing conditions holds:

• There is no injective function from X to the set of natural numbers.

• X is nonempty and every ω-sequence of elements of X fails to include at least one element of X. That is, X isnonempty and there is no surjective function from the natural numbers to X.

• The cardinality of X is neither finite nor equal to ℵ0 (aleph-null, the cardinality of the natural numbers).

• The set X has cardinality strictly greater than ℵ0 .

The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiomof choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

11.2 Properties

• If an uncountable set X is a subset of set Y, then Y is uncountable.

11.3 Examples

The best known example of an uncountable set is the set R of all real numbers; Cantor’s diagonal argument showsthat this set is uncountable. The diagonalization proof technique can also be used to show that several other sets areuncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of naturalnumbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or 2ℵ0 , or ℶ1

(beth-one).The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater thanzero but less than one (R has dimension one). This is an example of the following fact: any subset of R of Hausdorffdimension strictly greater than zero must be uncountable.

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42 CHAPTER 11. UNCOUNTABLE SET

Another example of an uncountable set is the set of all functions from R to R. This set is even “more uncountable”than R in the sense that the cardinality of this set is ℶ2 (beth-two), which is larger than ℶ1 .A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω1.The cardinality of Ω is denoted ℵ1 (aleph-one). It can be shown, using the axiom of choice, that ℵ1 is the smallestuncountable cardinal number. Thus either ℶ1 , the cardinality of the reals, is equal to ℵ1 or it is strictly larger. GeorgCantor was the first to propose the question of whether ℶ1 is equal to ℵ1 . In 1900, David Hilbert posed this questionas the first of his 23 problems. The statement that ℵ1 = ℶ1 is now called the continuum hypothesis and is known tobe independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice).

11.4 Without the axiom of choice

Without the axiom of choice, theremight exist cardinalities incomparable toℵ0 (namely, the cardinalities ofDedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above but not the fourth charac-terization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not wantto call them uncountable.If the axiom of choice holds, the following conditions on a cardinal κare equivalent:

• κ ≰ ℵ0;

• κ > ℵ0; and

• κ ≥ ℵ1 , where ℵ1 = |ω1| and ω1 is least initial ordinal greater than ω.

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriategeneralization of “uncountability” when the axiom fails. It may be best to avoid using the word in this case and specifywhich of these one means.

11.5 See also• Aleph number

• Beth number

• Injective function

• Natural number

11.6 References[1] Uncountably Infinite — from Wolfram MathWorld

• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

11.7 External links• Proof that R is uncountable

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Chapter 12

Universal set

For other uses, see Universal set (disambiguation).

In set theory, a universal set is a set which contains all objects, including itself.[1] In set theory as usually formulated,the conception of a universal set leads to a paradox (Russell’s paradox) and is consequently not allowed. However,some non-standard variants of set theory include a universal set.

12.1 Reasons for nonexistence

Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, donot allow for the existence of a universal set. Its existence would cause paradoxes which would make the theoryinconsistent.

12.1.1 Russell’s paradox

Russell’s paradox prevents the existence of a universal set in Zermelo–Fraenkel set theory and other set theories thatinclude Zermelo's axiom of comprehension. This axiom states that, for any formula φ(x) and any set A, there existsanother set

{x ∈ A | φ(x)}

that contains exactly those elements x of A that satisfyφ . If a universal set V existed and the axiom of comprehensioncould be applied to it, then there would also exist another set {x ∈ V | x ∈ x} , the set of all sets that do not containthemselves. However, as Bertrand Russell observed, this set is paradoxical. If it contains itself, then it should notcontain itself, and vice versa. For this reason, it cannot exist.

12.1.2 Cantor’s theorem

A second difficulty with the idea of a universal set concerns the power set of the set of all sets. Because this power setis a set of sets, it would automatically be a subset of the set of all sets, provided that both exist. However, this conflictswith Cantor’s theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality thanthe set itself.

12.2 Theories of universality

The difficulties associated with a universal set can be avoided either by using a variant of set theory in which theaxiom of comprehension is restricted in some way, or by using a universal object that is not considered to be a set.

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44 CHAPTER 12. UNIVERSAL SET

12.2.1 Restricted comprehension

There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V doesexist (and V ∈ V is true). In these theories, Zermelo’s axiom of comprehension does not hold in general, and theaxiom of comprehension of naive set theory is restricted in a different way. A set theory containing a universal set isnecessarily a non-well-founded set theory.The most widely studied set theory with a universal set is Willard Van Orman Quine’s New Foundations. AlonzoChurch and Arnold Oberschelp also published work on such set theories. Church speculated that his theory mightbe extended in a manner consistent with Quine’s,[2] but this is not possible for Oberschelp’s, since in it the singletonfunction is provably a set,[3] which leads immediately to paradox in New Foundations.[4] The most recent advancesin this area have been made by Randall Holmes who published an online draft version of the book Elementary SetTheory with a Universal Set in 2012.[5]

12.2.2 Universal objects that are not sets

Main article: Universe (mathematics)

The idea of a universal set seems intuitively desirable in the Zermelo–Fraenkel set theory, particularly because mostversions of this theory do allow the use of quantifiers over all sets (see universal quantifier). One way of allowingan object that behaves similarly to a universal set, without creating paradoxes, is to describe V and similar largecollections as proper classes rather than as sets. One difference between a universal set and a universal class is thatthe universal class does not contain itself, because proper classes cannot be elements of other classes. Russell’sparadox does not apply in these theories because the axiom of comprehension operates on sets, not on classes.The category of sets can also be considered to be a universal object that is, again, not itself a set. It has all sets aselements, and also includes arrows for all functions from one set to another. Again, it does not contain itself, becauseit is not itself a set.

12.3 Notes[1] Forster 1995 p. 1.

[2] Church 1974 p. 308. See also Forster 1995 p. 136 or 2001 p. 17.

[3] Oberschelp 1973 p. 40.

[4] Holmes 1998 p. 110.

[5] http://math.boisestate.edu/~{}holmes/

12.4 References• Alonzo Church (1974). “Set Theory with a Universal Set,” Proceedings of the Tarski Symposium. Proceedingsof Symposia in Pure Mathematics XXV, ed. L. Henkin, American Mathematical Society, pp. 297–308.

• T. E. Forster (1995). Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford Logic Guides31). Oxford University Press. ISBN 0-19-851477-8.

• T. E. Forster (2001). “Church’s Set Theory with a Universal Set.”• Bibliography: Set Theory with a Universal Set, originated by T. E. Forster and maintained by Randall Holmesat Boise State University.

• Randall Holmes (1998). Elementary Set theory with a Universal Set, volume 10 of the Cahiers du Centre deLogique, Academia, Louvain-la-Neuve (Belgium).

• Arnold Oberschelp (1973). “Set Theory over Classes,” Dissertationes Mathematicae 106.• WillardVanOrmanQuine (1937) “NewFoundations forMathematical Logic,”AmericanMathematicalMonthly44, pp. 70–80.

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12.5. EXTERNAL LINKS 45

12.5 External links• Weisstein, Eric W., “Universal Set”, MathWorld.

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46 CHAPTER 12. UNIVERSAL SET

12.6 Text and image sources, contributors, and licenses

12.6.1 Text• Countable set Source: https://en.wikipedia.org/wiki/Countable_set?oldid=665835456 Contributors: Damian Yerrick, AxelBoldt, Bryan

Derksen, Zundark, Xaonon, Danny, Oliverkroll, Kurt Jansson, Stevertigo, Patrick, Michael Hardy, Kku, Александър, Revolver, CharlesMatthews, Dysprosia, Hyacinth, Fibonacci, Head, Aleph4, Robbot, Romanm, MathMartin, Paul Murray, Ruakh, Tobias Bergemann,Giftlite, Everyking, Georgesawyer, Wyss, Simon Lacoste-Julien, Jorend, Hkpawn~enwiki, TheObtuseAngleOfDoom, Possession, Noisy,Rich Farmbrough, Pak21, Paul August, Gauge, Aranel, El C, PhilHibbs, Robotje, Jumbuck, Keenan Pepper, ABCD, Hu, Julioc, LOL,MattGiuca, Esben~enwiki, Graham87, Jetekus, Josh Parris, Salix alba, VKokielov, Kri, Chobot, Roboto de Ajvol, YurikBot, Taejo,Trovatore, Muu-karhu, Scs, Hirak 99, Lt-wiki-bot, Arthur Rubin, Reyk, Benandorsqueaks, Teply, Brentt, SmackBot, David Shear,Brick Thrower, Canthusus, Edonovan, Edgar181, Grokmoo, Xie Xiaolei, Silly rabbit, SEIBasaurus, Octahedron80, Javalenok, NYKevin,Matchups, SundarBot, Grover cleveland, Richard001, Bidabadi~enwiki, SashatoBot, Cronholm144, The Infidel, 16@r, Mets501, Newone,S0me l0ser, Martin Kozák, JRSpriggs, CRGreathouse, CBM, Mct mht, Gregbard, FilipeS, Thijs!bot, Colin Rowat, Magioladitis, Usien6,David Eppstein, MartinBot, Wdevauld, R'n'B, Ttwo, Qatter, Owlgorithm, Stokkink, Alyssa kat13, LokiClock, Rei-bot, Anonymous Dis-sident, Pieman93, Ilyaroz, Mike4ty4, SieBot, Caltas, Krishna.91, JackSchmidt, Unitvoice, Sunrise, Ken123BOT, Gigacephalus, Drag-onBot, Alexbot, Hans Adler, HumphreyW, Addbot, Topology Expert, Tomthecool, LaaknorBot, Super duper jimbo, LinkFA-Bot, Jarble,JakobVoss, Legobot, Luckas-bot, Yobot, AnomieBOT, Mihnea Maftei, Materialscientist, ArthurBot, Bdmy, Omnipaedista, Johnfranks,Worldrimroamer, Tkuvho, Phanxan, RedBot, Raiden09, EmausBot, QuantumOfHistory, Vishwaraj.anand00, Imcrazyaboutyou, AccessDenied, ClueBot NG, Wcherowi, Misshamid, Widr, Helpful Pixie Bot, Sinestar, Jim Sukwutput, Adityapanwarr, Dexbot, Sriharsh1234,Jochen Burghardt, Namespan, OliverBel, Matthew Kastor, Niceguy6, HKennethB and Anonymous: 127

• Empty product Source: https://en.wikipedia.org/wiki/Empty_product?oldid=655829085 Contributors: Damian Yerrick, Toby Bartels,Fubar Obfusco, Patrick, Michael Hardy, Komap, Paul A, Eric119, Ams80, Cyan, Revolver, Charles Matthews, WhisperToMe, McKay,Phil Boswell, Robbot, Fredrik, Altenmann, Henrygb, Wile E. Heresiarch, Giftlite, Paisa, ShaunMacPherson, Fropuff, Sundar, Cam-byses, Eequor, Fak119, Matt Crypto, CryptoDerk, Rlcantwell, Smyth, Paul August, Susvolans, Grick, Army1987, C S, La goutte depluie, Shreevatsa, Uncle G, Apokrif, MFH, Marudubshinki, Qwertyus, Jshadias, Chenxlee, Bubba73, Moskvax, Mathbot, Flashmor-bid, Trovatore, Nishantman, Ms2ger, WAS 4.250, Reyk, Bo Jacoby, SmackBot, InverseHypercube, Melchoir, Eskimbot, NoJoy, Oc-tahedron80, Javalenok, NYKevin, Daniel-Dane, Leland McInnes, Cybercobra, Daqu, MvH, EdC~enwiki, Happy-melon, Maxcantor,JRSpriggs, CBM, HenningThielemann, Cydebot, Headbomb, Dfrg.msc, RobHar, Ricardo sandoval, CommonsDelinker, Daniel5Ko, Os-sido, Steel1943, TXiKiBoT, Tom239, Anonymous Dissident, Dmcq, Thehotelambush, ClueBot, Watchduck, ChrisHodgesUK, Addbot,Ozob, Xario, ב ,.דניאל PV=nRT, Yobot, Citation bot, Charvest, D'ohBot, Citation bot 1, 777sms, Ebehn, Helpful Pixie Bot, Macofe andAnonymous: 43

• Empty set Source: https://en.wikipedia.org/wiki/Empty_set?oldid=666003197 Contributors: AxelBoldt, Lee Daniel Crocker, Uriyan,Bryan Derksen, Tarquin, Jeronimo, Andre Engels, XJaM, Christian List, Toby~enwiki, Toby Bartels, Ryguasu, Hephaestos, Patrick,Michael Hardy,MartinHarper, TakuyaMurata, Eric119, Den fjättrade ankan~enwiki, Andres, Evercat, Renamed user 4, CharlesMatthews,Berteun, Dcoetzee, David Latapie, Dysprosia, Jitse Niesen, Krithin, Hyacinth, Spikey, Jeanmichel~enwiki, Flockmeal, Phil Boswell,Robbot, Sanders muc, Peak, Romanm, Gandalf61, Henrygb, Wikibot, Pengo, Tobias Bergemann, Adam78, Tosha, Giftlite, Dbenbenn,Vfp15, BenFrantzDale, Herbee, Fropuff, MichaelHaeckel, Macrakis, Python eggs, Rdsmith4, Mike Rosoft, Brianjd, Mormegil, Guan-abot, Paul August, Spearhead, EmilJ, BrokenSegue, Nortexoid, 3mta3, Obradovic Goran, Jonathunder, ABCD, Sligocki, Dzhim, Itsmine,HenryLi, Hq3473, Angr, Isnow, Qwertyus, MarSch, Salix alba, Bubba73, ChongDae, Salvatore Ingala, Chobot, YurikBot, RussBot,Rsrikanth05, Trovatore, Ms2ger, Saric, EtherealPurple, GrinBot~enwiki, TomMorris, SmackBot, InverseHypercube, Melchoir, FlashSh-eridan, Ohnoitsjamie, Joefaust, SMP, J. Spencer, Octahedron80, Iit bpd1962, Tamfang, Cybercobra, Dreadstar, RandomP, Jon Awbrey,Jóna Þórunn, Lambiam, Jim.belk, Vanished user v8n3489h3tkjnsdkq30u3f, Loadmaster, Hvn0413, Mets501, EdC~enwiki, Joseph Solisin Australia, Spindled, James pic, Amalas, Philiprbrenan, CBM, Gregbard, Cydebot, Pais, Julian Mendez, Malleus Fatuorum, Epbr123,Nick Number, Escarbot, Sluzzelin, .anacondabot, David Eppstein, Ttwo, Maurice Carbonaro, Ian.thomson, It Is Me Here, Daniel5Ko,NewEnglandYankee, DavidCBryant, VolkovBot, Zanardm, Rei-bot, Anonymous Dissident, Andy Dingley, SieBot, Niv.sarig, ToePeu.bot,Randomblue, Niceguyedc, Wounder, Nosolution182, Versus22, Palnot, AmeliaElizabeth, Feinoha, American Eagle, ThisIsMyWikipedi-aName, LaaknorBot, AnnaFrance, Numbo3-bot, Zorrobot, Legobot, Luckas-bot, Yobot, Ciphers, Xqbot, Nasnema, , GrouchoBot,LucienBOT, Pinethicket, Kiefer.Wolfowitz, Abductive, Jauhienij, FoxBot, Lotje, LilyKitty, Woodsy dong peep, EmausBot, Sharlack-Hames, Ystory, ClueBot NG, Cntras, Rezabot, Helpful Pixie Bot, Michael.croghan, Langing, Ugncreative Usergname, JYBot, Kephir,Phinumu, Noyster, GeoffreyT2000, Skw27 and Anonymous: 82

• Fuzzy set Source: https://en.wikipedia.org/wiki/Fuzzy_set?oldid=669262232 Contributors: Zundark, Taw, Toby Bartels, Boleslav Bob-cik, Michael Hardy, MartinHarper, Ixfd64, Tgeorgescu, Александър, AugPi, Palfrey, Evercat, Charles Matthews, Markhurd, Furrykef,Hyacinth, Grendelkhan, VeryVerily, Robbot, Jaredwf, Peak, Giftlite, Jcobb, Duncharris, JasonQuinn, Phe, Urhixidur, Elwikipedista~enwiki,El C, Kwamikagami, R. S. Shaw, Pinar, Kusma, Joriki, Smmurphy, Ryan Reich, Salix alba, Mathbot, Predictor, YurikBot, Wavelength,Michael Slone, SpuriousQ, Gaius Cornelius, Srinivasasha, Supten, Jurriaan, Ml720834~enwiki, SmackBot, Hydrogen Iodide, CommanderKeane bot, Dreadstar, Rijkbenik, Bjankuloski06en~enwiki, Valepert, Elharo, JRSpriggs, George100, Paulmlieberman, CRGreathouse,Ksoileau, Gregbard, VashiDonsk, NotQuiteEXPComplete, Helgus, Nick Number, Abdel Hameed Nawar, Михајло Анђелковић, MER-C, Ty580, Bouktin, Magioladitis, MartinBot, Maurice Carbonaro, Gerla, DoorsAjar, Krzysiulek~enwiki, BotKung, LBehounek, In-formationSpace, Kilmer-san, VanishedUserABC, Cesarpermanente, ClueBot, Lukipuk, QYV, Pgallert, Multipundit, Addbot, Wirelessfriend, Legobot, Yobot, AnomieBOT, DemocraticLuntz, Riyad parvez, Pownuk, J JMesserly, Charvest, T2gurut2, Kierkkadon, Tin-ton5, Carel.jonkhout, FoxBot, Mjs1991, DixonDBot, The tree stump, WikitanvirBot, Matsievsky, Tijfo098, ChuispastonBot, ClueBotNG, Dezireh batist, Frietjes, Helpful Pixie Bot, StarryGrandma, Zbhsueh, Dannyeuu, Jcallega, Mark viking, Faizan, DangerouslyPersua-siveWriter, Atharkharal, IITHemant, Reddraggone9, RudiSeising, JMP EAX, Ffswontforget3 and Anonymous: 92

• Hereditarily finite set Source: https://en.wikipedia.org/wiki/Hereditarily_finite_set?oldid=622366694Contributors: TheAnome,MichaelHardy, Rp, Angela, Revolver, Dysprosia, Greenrd, Onebyone, Army1987, Oleg Alexandrov, R.e.b., SmackBot, Mhss, Dreadstar, JR-Spriggs, CRGreathouse, CBM, Ariel., CommonsDelinker, Watchduck, Addbot, Reconsider the static, ClueBot NG, Pastisch and Anony-mous: 7

• Infinite set Source: https://en.wikipedia.org/wiki/Infinite_set?oldid=659075298Contributors: TheAnome, Toby Bartels, DennisDaniels,Charles Matthews, David Shay, Bkell, Tobias Bergemann, Giftlite, Paul August, Rgdboer, Benji22210, Zerofoks, Salix alba, FlaBot,

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12.6. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 47

VKokielov, Chobot, DVdm, 4C~enwiki, Grubber, Trovatore, Maksim-e~enwiki, Addshore, Bidabadi~enwiki, Vina-iwbot~enwiki, Lam-biam, StevenPatrickFlynn, Bjankuloski06en~enwiki, Fell Collar, JRSpriggs, CRGreathouse, CBM, Escarbot, JAnDbot, Olaf, David Epp-stein, Ttwo, Maurice Carbonaro, Doug, DFRussia, Cliff, JP.Martin-Flatin, Alexbot, Hans Adler, Hatsoff, Addbot, Favonian, Luckas-bot, TaBOT-zerem, AnomieBOT, JackieBot, Materialscientist, VladimirReshetnikov, Erik9bot, Nicolas Perrault III, BenzolBot, Tku-vho, Pinethicket, SkyMachine, TobeBot, Beyond My Ken, Wgunther, Tommy2010, ZéroBot, Donner60, ClueBot NG, Widr, Juro2351,Magic6ball, Sauood07, YFdyh-bot, Blackbombchu, K9re11, Centralpanic and Anonymous: 37

• Recursive set Source: https://en.wikipedia.org/wiki/Recursive_set?oldid=662150948 Contributors: Michael Hardy, Docu, Ehn, Hy-acinth, Robbot, MathMartin, Ojigiri~enwiki, Saforrest, Tobias Bergemann, Filemon, Giftlite, Peruvianllama, Khalid hassani, Vivero~enwiki,Satyadev, Obradovic Goran, Arthena, Salix alba, NekoDaemon, BMF81, YurikBot, Hairy Dude, TheGrappler, Trovatore, R.e.s., ArthurRubin, Maksim-e~enwiki, Eskimbot, Mhss, Benkovsky, Adibob, Viebel, Feradz, Mets501, JRSpriggs, Ylloh, CBM, Gregbard, Cydebot,Thijs!bot, Ttwo, CBM2, Justin W Smith, DumZiBoT, BodhisattvaBot, Addbot, Yobot, Pcap, Almabot, VladimirReshetnikov, ZéroBot,AvicAWB, Fesharakif and Anonymous: 13

• Subset Source: https://en.wikipedia.org/wiki/Subset?oldid=670318766 Contributors: Damian Yerrick, AxelBoldt, Youssefsan, XJaM,Toby Bartels, StefanRybo~enwiki, Edward, Patrick, TeunSpaans, Michael Hardy, Wshun, Booyabazooka, Ellywa, Oddegg, Andres,Charles Matthews, Timwi, Hyacinth, Finlay McWalter, Robbot, Romanm, Bkell, 75th Trombone, Tobias Bergemann, Tosha, Giftlite,Fropuff, Waltpohl, Macrakis, Tyler McHenry, SatyrEyes, Rgrg, Vivacissamamente, Mormegil, EugeneZelenko, Noisy, Deh, Paul Au-gust, Engmark, Spoon!, SpeedyGonsales, Obradovic Goran, Nsaa, Jumbuck, Raboof, ABCD, Sligocki, Mac Davis, Aquae, LFaraone,Chamaeleon, Firsfron, Isnow, Salix alba, VKokielov, Mathbot, Harmil, BMF81, Chobot, Roboto de Ajvol, YurikBot, Alpt, Dmharvey,KSmrq, NawlinWiki, Trovatore, Nick, Szhaider, Wasseralm, Sardanaphalus, Jacek Kendysz, BiT, Gilliam, Buck Mulligan, SMP, Or-angeDog, Bob K, Dreadstar, Bjankuloski06en~enwiki, Loadmaster, Vedexent, Amitch, Madmath789, Newone, CBM, Jokes Free4Me,345Kai, SuperMidget, Gregbard, WillowW, MC10, Thijs!bot, Headbomb, Marek69, RobHar, WikiSlasher, Salgueiro~enwiki, JAnDbot,.anacondabot, Pixel ;-), Pawl Kennedy, Emw, ANONYMOUS COWARD0xC0DE, RaitisMath, JCraw, Tgeairn, Ttwo, Maurice Car-bonaro, Acalamari, Gombang, NewEnglandYankee, Liatd41, VolkovBot, CSumit, Deleet, Rei-bot, AnonymousDissident, James.Spudeman,PaulTanenbaum, InformationSpace, Falcon8765, AlleborgoBot, P3d4nt, NHRHS2010, Garde, Paolo.dL, OKBot, Brennie8, Jons63,Loren.wilton, ClueBot, GorillaWarfare, PipepBot, The Thing That Should Not Be, DragonBot, Watchduck, Hans Adler, Computer97,Noosentaal, Versus22, PCHS-NJROTC, Andrew.Flock, Reverb123, Addbot, , Fyrael, PranksterTurtle, Numbo3-bot, Zorrobot, Jar-ble, JakobVoss, Luckas-bot, Yobot, Synchronism, AnomieBOT, Jim1138, Materialscientist, Citation bot, Martnym, NFD9001, Char-vest, 78.26, XQYZ, Egmontbot, Rapsar, HRoestBot, Suffusion of Yellow, Agent Smith (The Matrix), RenamedUser01302013, ZéroBot,Alexey.kudinkin, Chharvey, Quondum, Chewings72, 28bot, ClueBot NG, Wcherowi, Matthiaspaul, Bethre, Mesoderm, O.Koslowski,AwamerT, Minsbot, Pratyya Ghosh, YFdyh-bot, Ldfleur, ChalkboardCowboy, Saehry, Stephan Kulla, , Ilya23Ezhov, Sandshark23,Quenhitran, Neemasri, Prince Gull, Maranuel123, Alterseemann, Rahulmr.17 and Anonymous: 181

• Tarski–Grothendieck set theory Source: https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory?oldid=649863373Contributors: Dwheeler, Charles Matthews, R3m0t, Tobias Bergemann, Langec, RayBirks, EmilJ, Cherlin, Mdd, JosefUrban, Nmegill,Trovatore, SmackBot, JRSpriggs, CRGreathouse, CBM, Sam Staton, Jessealama, Wasell, Daniel5Ko, JohnBlackburne, JP.Martin-Flatin,DumZiBoT, Addbot, Yobot, EmausBot, UniversumExNihilo, Brad7777, K9re11 and Anonymous: 16

• Transitive set Source: https://en.wikipedia.org/wiki/Transitive_set?oldid=659275410 Contributors: Edward, Charles Matthews, JitseNiesen, Tobias Bergemann, Lethe, EmilJ, Oleg Alexandrov, Salix alba, Arthur Rubin, Mhss, Keithdunwoody, JRSpriggs, Vaughan Pratt,CBM, Gregbard, Roches, Ttwo, Franklin.vp, Addbot, Barak Sh, Luckas-bot, Xqbot, Erik9bot, Tkuvho, Wikielwikingo, EmausBot,ZéroBot, Deltahedron, Pastisch and Anonymous: 13

• Uncountable set Source: https://en.wikipedia.org/wiki/Uncountable_set?oldid=664962517Contributors: AxelBoldt, Tarquin, AstroNomer~enwiki,Taw, Toby Bartels, PierreAbbat, Patrick, Michael Hardy, Dominus, Kevin Baas, Revolver, Charles Matthews, Dysprosia, Hyacinth, Fi-bonacci, Aleph4, Robbot, Tobias Bergemann, Giftlite, Mshonle~enwiki, Fropuff, Noisy, Crunchy Frog, Func, Keenan Pepper, OlegAlexandrov, Graham87, Island, Salix alba, FlaBot, Margosbot~enwiki, YurikBot, Piet Delport, Gaius Cornelius, Trovatore, Scs, Bota47,Arthur Rubin, Naught101, SmackBot, Bh3u4m, Bananabruno, SundarBot, Dreadstar, Germandemat, Loadmaster, Mets501, Limaner,Stephen B Streater, JRSpriggs, CRGreathouse, CBM, Gregbard, Thijs!bot, Dugwiki, Salgueiro~enwiki, JAnDbot, .anacondabot, Ttwo,Qatter, KarenJo90, SieBot, Phe-bot, ClueBot, Canopus1, DumZiBoT,Addbot, Yobot, Omnipaedista, BenzolBot, FoxBot, Vishwaraj.anand00,Mark viking, ILLUSION-ZONE and Anonymous: 33

• Universal set Source: https://en.wikipedia.org/wiki/Universal_set?oldid=667869756 Contributors: Awaterl, Patrick, Charles Matthews,Dysprosia, Hyacinth, Paul August, Jumbuck, Gary, Salix alba, Chobot, Hairy Dude, SmackBot, Incnis Mrsi, FlashSheridan, Gilliam,Lambiam, AndriusKulikauskas, Newone, CBM, User6985, Cydebot, LookingGlass, David Eppstein, Ttwo, VolkovBot, Anonymous Dis-sident, SieBot, ToePeu.bot, Oxymoron83, Cliff, Addbot, Neodop, Download, Dimitris, Yobot, Shlakoblock, Citation bot, Xqbot, Amaury,FrescoBot, Aikidesigns, Petrb, Wcherowi, Jochen Burghardt, Vivianthayil, Smortypi, Blackbombchu, TerryAlex and Anonymous: 24

12.6.2 Images• File:Aplicación_2_inyectiva_sobreyectiva02.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Aplicaci%C3%B3n_

2_inyectiva_sobreyectiva02.svg License: Public domain Contributors: Own work Original artist: HiTe• File:Empty_set.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/aa/Empty_set.svg License: Public domain Contribu-tors: Own work Original artist: Octahedron80

• File:Loupe_light.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0c/Loupe_light.svg License: Public domain Contrib-utors: modified version of <a href='//commons.wikimedia.org/wiki/File:Gnome-searchtool.svg' class='image'><img alt='Gnome-searchtool.svg'src='https://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gnome-searchtool.svg/50px-Gnome-searchtool.svg.png' width='50'height='50' srcset='https://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gnome-searchtool.svg/75px-Gnome-searchtool.svg.png1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gnome-searchtool.svg/100px-Gnome-searchtool.svg.png 2x' data-file-width='60' data-file-height='60' /></a> Original artist: Watchduck (a.k.a. Tilman Piesk)

• File:Nested_set_V4.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1b/Nested_set_V4.svg License: Public domainContributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)

• File:Nullset.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6a/Nullset.svg License: CC BY-SA 3.0 Contributors: Ownwork Original artist: Hugo Férée

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48 CHAPTER 12. UNIVERSAL SET

• File:Pairing_natural.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6f/Pairing_natural.svg License: CC-BY-SA-3.0Contributors: Own work Original artist: Cronholm144

• File:PolygonsSet_EN.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/de/PolygonsSet_EN.svg License: CC0 Contrib-utors: File:PolygonsSet.svg Original artist: File:PolygonsSet.svg: kismalac

• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

• File:Subset_with_expansion.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Subset_with_expansion.svg License:CC-BY-SA-3.0 Contributors: Own work Original artist: User:J.Spudeman~{}commonswiki

• File:Venn_A_intersect_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6d/Venn_A_intersect_B.svg License: Pub-lic domain Contributors: Own work Original artist: Cepheus

• File:Venn_A_subset_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b0/Venn_A_subset_B.svg License: Public do-main Contributors: Own work Original artist: User:Booyabazooka

• File:Wiktionary-logo-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License: Publicdomain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk · contribs),based on original logo tossed together by Brion Vibber

12.6.3 Content license• Creative Commons Attribution-Share Alike 3.0


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