Chapter I Set Theorydoud/Transition/Sets.pdf · 2019-06-17 · Chapter I Set Theory There is surely...

Chapter I

Set Theory

There is surely a piece of divinity in us, something that was before the elements, and

owes no homage unto the sun. Sir Thomas Browne

One of the benefits of mathematics comes from its ability to express a lot ofinformation in very few symbols. Take a moment to consider the expression

d

dθsin(θ).

It encapsulates a large amount of information. The notation sin(θ) represents, fora right triangle with angle θ, the ratio of the opposite side to the hypotenuse. Thedifferential operator d/dθ represents a limit, corresponding to a tangent line, and soforth.

Similarly, sets are a convenient way to express a large amount of information.They give us a language we will find convenient in which to do mathematics. This isno accident, as much of modern mathematics can be expressed in terms of sets.

1

2 CHAPTER I. SET THEORY

1 Sets, subsets, and set operations

1.A What is a set?

A set is simply a collection of objects. The objects in the set are called the elements.We often write down a set by listing its elements. For instance, the set S = {1, 2, 3}

has three elements. Those elements are 1, 2, and 3. There is a special symbol, ∈,that we use to express the idea that an element belongs to a set. For instance, wewrite 1 ∈ S to mean that “1 is an element of S.”

For the set S = {1, 2, 3}, we have 1 ∈ S, 2 ∈ S, and 3 ∈ S. We can write thismore quickly as: 1, 2, 3 ∈ S. We can express the fact that 4 is not an element of Sby writing 4 /∈ S.

Example 1.1. Let P be the set {16,−5, 2, 6, 9}. Is 6 ∈ P? Yes! Is 5 ∈ P? No, sowe write 5 /∈ P . △

The order of the elements in a set does not matter, so we could have writtenS = {1, 2, 3} as S = {1, 3, 2}, or as S = {3, 2, 1}. If an element is repeated in a set,we do not count the multiplicity. Thus {1, 2, 3, 1} is the same set as S = {1, 2, 3}.We say that two sets are equal when they have exactly the same elements.

Not all sets consist of numbers. For instance T = {a, b, c, d} is a set whose elementsare the letters a, b, c, d. Sets may have words, names, symbols, and even other sets aselements.

Example 1.2. Suppose we want to form the set of Jesus’ original twelve apostles.This would be the set

Apostles = {Peter, James, John the beloved, . . . , Judas Iscariot}.We put the 3 dots in the middle to express the fact that there are more elementswhich we have not listed (perhaps to save time and space). △

The Last Supper, ca. 1520, by Giovanni Pietro Rizzoli.

The next example is a set with another set as an element.

1. SETS, SUBSETS, AND SET OPERATIONS 3

Example 1.3. Let S = {1, 5, {4, 6}, 3}. This set has four elements. We have1, 5, 3, {4, 6} ∈ S, but 4 /∈ S. However, 4 ∈ {4, 6} and {4, 6} ∈ S. △

It can be confusing when sets are elements of other sets. You might ask whymathematicians would allow such confusion! It turns out that this is a very usefulthing to allow; just like when moving, the moving truck (a big box) has boxes insideof it, each containing other things.

Advice 1.4. You can think of sets as boxes with objects inside. So we couldview the set {1, 5, {4, 6}, 3} from the previous example as the following box,which contains another box:

1 5 4 6 3

Figure 1.4: A box with a box inside, each containing some numbers.

1.B Naming sets

We’ve seen that capitalized Roman letters can be used to give names to sets. Somesets are used so often that they are represented by special symbols. Here are a coupleof examples.

• The set of natural numbers is the set

N = {1, 2, 3, . . .}.

This is the first example we’ve given of an infinite set, i.e., a set with infinitelymany elements.

• The set of integers is

Z = {. . . ,−3,−2,−1, 0, 1, 2, 3, 4, . . .}.

The dots represent the fact that we are leaving elements unwritten in bothdirections. We use the fancy letter “Z” because the word “integer” in Germanis “Zahlen.”

Some sets are constructed using rules. For example, the set of even integers canbe written as

{. . . ,−4,−2, 0, 2, 4, . . .}but could also be written in the following ways:

{2x : x ∈ Z}(1.5)

{x ∈ Z : x is an even integer}(1.6)

{x : x = 2y for some y ∈ Z}.(1.7)


We read the colon as “such that,” so (1.5) is read as “the set of elements of the form2x such that x is an integer.” Writing sets with a colon is called set-builder notation.Notice that

{x ∈ Z : 2x+ 1}doesn’t make any sense, since “2x+ 1” is not a condition on x.

Here are a few more examples. The set of prime numbers is

{2, 3, 5, 7, 11, 13, . . .} = {x ∈ N : x is prime}.

Similarly, Apostles = {x : x was one of the original 12 apostles of Jesus}.With set-builder notation, we can list a few more very important sets.• The set of rational numbers is

Q = {a/b : a, b ∈ Z, b 6= 0}.

Note that there is no problem with the fact that different fractions can representthe same rational number, such as 1/2 = 2/4. Repetitions do not matter insets. We will occasionally need the fact that we can always write a rationalnumber as a fraction a/b in lowest terms: i.e., so that a and b have no commonfactor larger than 1. We will prove this in Section 18 (see Exercise 18.3).

• The set of real numbers is

R = {x : x has a decimal expansion}.

So we have π = 3.14159 . . . ∈ R, 3 = 3.00000 . . . ∈ R, and√2 ∈ R. Later in

this book we will prove√2 /∈ Q.

• The set of complex numbers is

C = {a+ bi : a, b ∈ R, i2 = −1}.

Is 3 a complex number? Yes, because we can take a = 3 and b = 0. So we have3 ∈ N, 3 ∈ Z, 3 ∈ Q, 3 ∈ R, and 3 ∈ C!

Example 1.8. Which of the named sets does π = 3.14159 . . . belong to? We haveπ ∈ R and π ∈ C. On the other hand, since 3 < π < 4 we have π /∈ N and π /∈ Z. Itis true, but much harder to show, that π /∈ Q. △

There is one more set we will give a special name.• The empty set is the set with no elements. We write it as ∅ = { }.

Warning 1.9. The empty set is not nothing. It has no elements, but the emptyset is something. Namely, it is “the set with nothing in it”.

Thinking in terms of boxes, we can think of the empty set as an empty box.The box is something even if it has nothing in it.

The symbol ∅ does not mean nothing. It means { }.


Example 1.10. Sometimes we want the empty set to be an element of a set. Forinstance, we might take

S = {∅}.The set S has a single element, namely ∅. We could also write S = {{ }}. In terms ofboxes, S is the box containing an empty box. Note that not all sets have the emptyset as an element. △

A box with an empty box inside, representing {∅}.

1.C Subsets

In many activities in life we don’t focus on all the elements of a set, but rather onsubcollections. To give just a few examples:

• The set of all phone numbers is too large for most of us to handle. The subcol-lection of phone numbers of our personal contacts is much more manageable.

• If we formed the set of all books ever published in the world, this set would bevery large (but still finite!). However, the subcollection of books we have readis much smaller.

• If we want to count how many socks we own, we could use elements of theintegers Z, but since we cannot own a negative number of socks, a more naturalset to use would be the subcollection of nonnegative integers

Z≥0 = {0, 1, 2, 3, . . .}.

A subcollection of a set is called a subset. When A is a subset of B we writeA ⊆ B and if it is not a subset we write A * B. There are a couple different ways tothink about the concept of A being a subset of B.

Option 1: To check that A is a subset of B, we check that every element of A alsobelongs to B.

Example 1.11. (1) Let A = {1, 5, 6} and B = {1, 5, 6, 7, 8}. Is A a subset of B?Yes, because we can check that each of A’s three elements, 1, 5, and 6, belongsto B.

(2) Let A = {6, 7/3, 9, π} and B = {1, 2, 6, 7/3, π, 10}. Is A ⊆ B? No, because9 ∈ A but 9 /∈ B. So we write A * B.

(3) Let A = N and B = Z. Is A ⊆ B? Yes, every natural number is an integer.(4) Is Z a subset of N? No, because Z has the element −1, which doesn’t belong

to N. △


Option 2: To check that A is a subset of B, we check that we can form A by throwingout some of the elements of B.

Example 1.12. (1) Let A = {1, 5, 6} and B = {1, 5, 6, 7, 8}. Is A a subset of B?Yes, because we can throw away 7, 8 from B to get A.

(2) Let A = {6, 7/3, 9, π} and B = {1, 2, 6, 7/3, π, 10}. Is A a subset of B? No,because as we throw away elements of B, we can never get 9 inside.

(3) Let A = N and B = Z. Is A a subset of B? Yes, because we can throw awaythe negative integers and 0 to get the natural numbers.

(4) Is Z a subset of N? No, because we cannot get −1 by throwing away elementsfrom N. △

When we write A ⊆ B, the little line segment at the bottom of “⊆” means thatthere is possible equality. (Just like x ≤ y means that x is less than or equal to y.)Sometimes we do not want to allow equality. We use the following terminology inthis case.

Definition 1.13. If A ⊆ S and A 6= S, we say that A is a proper subset of S,and we write A ( S.

Note that the symbol ( is different from *. If A ( B, then A is a subset of Bthat is not equal to B, while if A * B, then A is not a subset of B.

Example 1.14. We have {1, 2} ( {1, 2, 3}. Of course {1, 2} ⊆ {1, 2, 3} is alsotrue. △

Warning 1.15. Some authors use ⊂ instead of ⊆. Other authors use ⊂ insteadof (. Thus, there can be a lot of confusion about what ⊂ means, which is onereason why we will avoid that notation in this book!

Warning 1.16. Many students learning about subsets get confused about thedifference between being an element and being a subset. Consider your musiclibrary as a set. The elements are the individual songs. Playlists, which arecollections of some of the songs, are subsets of your library.

Example 1.17. The elements of C are complex numbers like 3+6i or −2.7−5.9i. Thesubsets of C are sets of complex numbers like {5.4−7.3i, 9+0i,−2.671+9.359i}. △Example 1.18. (1) Let T = {1, 2, 3, 4, 5}. Is 2 an element or a subset of T? It

is an element, since it lives inside T . It is not a subset, since it isn’t a set ofelements of T .

(2) Let U = {−5, 6, 7, 3}. Is {6} an element or a subset of U? It is not an elementof U , since the set {6} isn’t in its list of elements. It is a subset because it is abox whose elements come from U .


(3) Let X = {{6}, {7, 8}, {5, 8}}. Is 7 an element or a subset of X? Neither! It isnot one of the three elements listed in X, and it is not a box of elements in Xeither.Is {7, 8} an element or a subset of X? It is an element, since it is one of thethree listed elements. It is not a subset, even though it is a box, since it haselements which don’t belong to X.

(4) Let Y = {5, {5}}. Is {5} an element or a subset of Y ? It is both! It is anelement, since it is the second element listed inside Y . It is also a subset of Y ,since it is a box containing the first element of Y . △

It can be useful to construct sets satisfying certain properties in relation to oneanother. In the following example we show how this can be done.

Example 1.19. We will find three sets A,B,C satisfying the following conditions:(1) A ⊆ B,(2) A ∈ C, and(3) C ⊆ B with C 6= B (i.e., C ( B).One method to solve this problem is to start with the simplest sets possible and

modify them as needed. So let’s start with

A = { }, B = { }, C = { }.We see that condition (1) is fulfilled, but condition (2) is not. To force condition (2)to be true, we must make A an element of C. Thus, our new sets are

A = { }, B = { }, C = {A}.Condition (1) still holds, and condition (2) is now true. However, condition (3) doesn’thold. To make (3) true, we need B to have all the elements of C and at least onemore. So we take

A = { }, B = {A, 1}, C = {A}.We double-check that all of the conditions hold (which they do), and so we have ourfinal answer. △

1.D Cardinality

The number of elements of a set is called its cardinality. For instance, the set S ={1, 2, 3} has 3 elements. We write |S| = 3 to denote that S has cardinality 3. Notethat |∅| = 0 but |{∅}| = 1. A set is finite if its cardinality is either 0 or a naturalnumber, and it is infinite otherwise. In a later section in the book, we will talk abouta better way to define cardinality for infinite sets.

Example 1.20. If T = {5, {6, 7, 8}, {3}, 0, ∅}, the cardinality is |T | = 5. △In mathematics, we sometimes use the same symbols for two different things. The

meaning of the symbols must be deduced from their context. For instance, if we write|−3.392| this is certainly not the cardinality of a set, but instead is probably referringto the absolute value of a number. In the next example, we use | · | in two differentways.


Example 1.21. If T = {x ∈ Z : |x| < 4}, what is |T |?(Hint: It is bigger than 4.) △

1.E Power sets

In this section we define the power set and give some examples.

Definition 1.22. Let S be a set. The power set of S is the new set P(S) whoseelements are the subsets of S. In other words, A ∈ P(S) exactly when A ⊆ S.

The next example determines the power set of a small set S.

Example 1.23. Can we list all of the subsets of S = {1, 2, 3}? If we think aboutsubsets as “boxes containing only elements of S”, we just have to list all possibilities.They are as follows:

P(S) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

Why is the empty set one of the subsets? Is it really a box containing only elementsof S? Yes, its elements (there are none!) all belong to S. Thinking about it in termsof “throwing away” elements of S, we threw all of them away.

Why is S ⊆ S? Because S is a box containing only elements of S. Thinking interms of “throwing away” elements, we threw away none of the elements. △

If S is a finite set, we can determine the size of the power set |P(S)| from |S|.

Theorem 1.24. If |S| = n, then |P(S)| = 2n.

Here is a sketch of why this is true. To form a subset of S, for each element in S wechoose to keep or throw away that element. Thus, there are 2 choices for each element.Since there are n elements, this gives 2n options.

Example 1.25. For the set S = {1, 2, 3} we have |S| = 3. Thus the power set hascardinality |P(S)| = 23 = 8. This is exactly the number of elements we listed inExample 1.23. △Example 1.26. How may elements will the power set of U = {1, ∅} have? The setU has two elements, so there should be 22 = 4 subsets. They can be listed as:

P(U) = {∅, {1}, {∅}, U}.

Which of these are proper? (All of them except U itself.) △Example 1.27. List three elements of P(N), each having different cardinality, andone being infinite.

Here is one possible answer: {1, 7}, {67, 193, 91948}, and {2, 4, 6, 8, 10, . . .}. Thereare many other correct choices. △


1.F Unions and intersections

There are multiple ways to modify sets. When there are two sets S and T , we canput them together to form a new set called the union, and we write

S ∪ T = {x : x ∈ S or x ∈ T}.

This is the set of elements which belong to S or T or both of them. (When we use theword “or” in this book, we will almost always use the inclusive meaning.) Pictorially,we can view this set using a Venn diagram as follows.

S T

Figure 1.28: The union of S and T .

Similarly, given two sets S and T we can form the set of elements that belong toboth of them, called the intersection, and we write

S ∩ T = {x : x ∈ S and x ∈ T}.

The Venn diagram is the following.

S T

Figure 1.29: The intersection of S and T .

Example 1.30. Let A = {1, 6, 17, 35} and B = {1, 5, 11, 17}. Then

A ∪B = {1, 5, 6, 11, 17, 35}, A ∩B = {1, 17}. △

Example 1.31. Find sets P,Q with |P | = 7, |Q| = 9, and |P ∩ Q| = 5. How big is|P ∪Q|?


We start by letting P be the easiest possible set with 7 elements, namely P ={1, 2, 3, 4, 5, 6, 7}. Since Q must share 5 of these elements, but have 9 elements total,we could write Q = {1, 2, 3, 4, 5, 8, 9, 10, 11}.

For the example we constructed, we have P ∪ Q = {x ∈ N : x ≤ 11}, so|P∪Q| = 11. If we chose other sets P and Q, could |P∪Q| be different? (Answer: No.The given numbers determine the cardinality of each piece in the Venn diagram.) △

1.G Complements and differences

Let S and T be sets. The difference of T and S is

T − S = {x : x ∈ T and x /∈ S}.

The Venn diagram is as follows.

S T

Figure 1.32: The difference of T and S.

Example 1.33. Let S = {1, 2, 3, 4, 5, 6, 7} and T = {6, 7, 8, 9}. We find T − S is theset

T − S = {8, 9}.

Notice that we do not need to worry about those elements of S which do not belongto T . We only have to take away the part the two sets share. So T −S = T − (S∩T ).

Also notice that S − T = {1, 2, 3, 4, 5} is different from T − S. △

Example 1.34. Let A and B be sets. Assume |A| = 16 and |B| = 9. If |A∩B| = 2,what are |A− B| and |B − A|?

There are only two elements that A andB share, thus |A−B| = 14 and |B−A| = 7.

Can you now figure out |A ∪B|? (Hint: Draw the Venn diagram.) △

Occasionally we will be working inside some set U , which we think of as theuniversal set for the problem at hand. For instance, when solving quadratic equations,such as x2 − x+ 2 = 0, your universal set might be the complex numbers C.

Given a subset S of the universal set U , we write S = U − S, and call this thecomplement of S (in the universal set U). The Venn diagram follows.


S

U

Figure 1.35: The complement of a set S inside a universal set U .

Example 1.36. Let U = N and let P = {2, 3, 5, 7, . . .} be the set of primes.What is P? This is the set of composite numbers and 1, or in other words P ={1, 4, 6, 8, 9, . . .}. △

1.H Exercises

Exercise 1.1. Each of the following sets is written in set-builder notation. Write theset by listing its elements. Also state the cardinality of each set.(a) S1 = {n ∈ N : 5 < |n| < 11}.(b) S2 = {n ∈ Z : 5 < |n| < 11}.(c) S3 = {x ∈ R : x2 + 2 = 0}.(d) S4 = {x ∈ C : x2 + 2 = 0}.(e) S5 = {t ∈ Z : t5 < 1000}. (This one is slightly tricky.)

Exercise 1.2. Rewrite each of the following sets in the form{x ∈ S : some property on x},

just as we did in (1.6) above, by finding an appropriate property.(a) A1 = {1, 3, 5, 7, 9, . . .} where S = N.(b) A2 = {1, 8, 27, 64, . . .} where S = N.(c) A3 = {−1, 0} where S = {−1, 0, 1}.

Exercise 1.3. Write the following sets in set-builder notation.(a) A = {. . . ,−10,−5, 0, 5, 10, 15, . . .}.(b) B = {. . . ,−7,−2, 3, 8, 13, 18, . . .}.(c) C = {1, 16, 81, 256, . . .}.(d) D = {. . . , 1/4, 1/2, 1, 2, 4, 8, 16, . . .}.

Exercise 1.4. Give specific examples of sets A, B, and C satisfying the followingconditions (in each part, separately):(a) A ∈ B, B ∈ C, and A /∈ C.(b) A ∈ B, B ⊆ C, and A * C.(c) A ( B, B ∈ C, and A ∈ C.(d) A ∩B ⊆ C, A * C, and B * C.(e) A ∩ C = ∅, A ⊆ B, |B ∩ C| = 3.


Exercise 1.5. Let A = {1, 2}. Find P(A), and then find P(P(A)). What are thecardinalities of these three sets?

Exercise 1.6. Let a, b ∈ R with a < b. The closed interval [a, b] is the set {x ∈ R :a ≤ x ≤ b}. Similarly, the open interval (a, b) is the set {x ∈ R : a < x < b}. LetP = [3, 7], Q = [7, 9] and R = [−3, 8]. Give simple descriptions of the following sets.(a) P ∩Q.(b) P ∪Q.(c) P −Q.(d) Q− P .(e) (R ∩ P )−Q.(f) (P ∪Q) ∩R.(g) P ∪ (Q ∩R).

Exercise 1.7. Consider the following blank Venn diagram for the three sets A,B,C.

B C

A

For each of the following sets, copy the Venn diagram above, and then shade in thenamed region:(a) A− (B ∩ C).(b) A− (B − C).(c) B − (A− C).(d) (B ∩ C) ∩ (B ∪ A).(e) (A− B) ∪ (A− C).

Exercise 1.8. Two sets S, T are disjoint if they share no elements. In other wordsS ∩ T = ∅. Which of the following sets are disjoint? Give reasons.(a) The set of odd integers and the set of even integers.(b) The natural numbers and the complex numbers.(c) The prime numbers and the composite numbers.(d) The rational numbers and the irrational numbers (i.e., real numbers which are

not rational).

Exercise 1.9. Find some universal set U and subsets S, T ⊆ U , such that |S−T | = 3,|T − S| = 1, |S ∪ T | = 6, and |S| = 2. (Write each of U , S, and T by listing theirelements.)

2. PRODUCTS OF SETS AND INDEXED SETS 13

2 Products of sets and indexed sets

2.A Cartesian products

Sets are unordered lists of elements. There are situations where order matters. Forinstance, you probably don’t want to put your shoes on before your socks. To give amore mathematical example, if we square a number and then take its cosine, that isnot the same as first taking the cosine and then squaring:

cos(x2) 6= (cos(x))2.

There are other situations where we want to keep things ordered. We will write (x, y)for the ordered pair where x occurs first and y occurs second. Thus (x, y) 6= (y, x)even though {x, y} = {y, x}. Also, an element can be repeated in an ordered list,such as (1, 1), while sets do not count repetitions.

There is a very nice notation for sets of ordered pairs.

Definition 2.1. Let S and T be two sets. The Cartesian product of these setsis the new set

S × T = {(s, t) : s ∈ S, t ∈ T}.This is the set of all ordered pairs such that the first entry comes from S andthe second entry comes from T . We will often refer to S×T just as the productof S and T .

We will now give an example of how to find simple Cartesian products.

Example 2.2. Let S = {1, 2, 3} and T = {1, 2}. What is S × T? It is the set{(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2)}. Notice that 3 can occur as a first coordinatesince 3 ∈ S, but not as a second coordinate since 3 /∈ T .

While the order matters inside an ordered pair, we could have listed the elementsof S × T in a different order since S × T is itself just a set (and order is irrelevant insets). So we could have written

S × T = {(1, 2), (2, 2), (3, 1), (1, 1), (2, 1), (3, 2)}.

However,S × T 6= T × S = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}. △

You might notice that in the previous example we have |S×T | = 6 = 3·2 = |S|·|T |.This is not an accident. In fact, the following is true, although we do not as yet havethe tools to prove it.

Proposition 2.3. Let A and B be finite sets, with |A| = m and |B| = n. ThenA× B is a finite set, with |A× B| = mn.


Sets do not need to be finite in order to act as components in products.

Example 2.4. Let A = N and B = {0, 1}. What are the elements of A× B? Theyare

A× B = {(1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1), . . .} = {(n, 0), (n, 1) : n ∈ N}.

Is A × B the same set as B × A? No, they have different elements. For instance,(1, 0) ∈ A× B, but (1, 0) /∈ B × A since 0 /∈ A. △

In each of the previous examples, we took the Cartesian product of two different

sets. If we take the product of a set with itself, we sometimes write A2 = A×A. Thefollowing example is one of the most useful products of a set with itself.

Example 2.5. The set R2 = R×R is called the Cartesian plane. We view elementsin this set as points {(x, y) : x, y ∈ R}.

x

y

The Cartesian plane, R× R

The set S × T in Example 2.2 is a subset of R× R. We can graph it as follows:

x

y

{1, 2, 3} × {1, 2}


Similarly, the set A× B from Example 2.4 is graphed as:

x

y

N× {0, 1}

· · ·

· · ·

We can now describe more complicated sets. For instance

{(x, y) ∈ R× R : y = 3x+ 1}

is a line. The set R2 − {(0, 0)} is the punctured plane (the plane with the originremoved). Can you describe a simple parabola? △

The following example addresses the question: “What do we do if one of the setshas no elements?”

Example 2.6. We determine {1, 2, 3} × ∅. Elements of this set are ordered pairs ofthe form (a, b), with a ∈ {1, 2, 3} and b ∈ ∅. Thus, there are no possible choices forb, and so {1, 2, 3} × ∅ = ∅. Note that 3 · 0 = 0, so Proposition 2.3 works in this casetoo. △

Just as with ordered pairs, we can form the set of ordered triples

A× B × C = {(a, b, c) : a ∈ A, b ∈ B, c ∈ C}.

We can similarly form ordered quadruples, ordered quintuples, and so forth. Thenext subsection will give us the tools necessary to talk about even more complicatedconstructions.

2.B Indices

When we have a large number of sets, rather than writing them using different lettersof the alphabet

A,B,C,D, . . . , Z

it can be easier to use subscripts

A1, A2, A3, A4, . . . , A26.

This notation is extremely powerful for the following reasons:


• The notation tells us how many sets we are working with, using a small numberof symbols. For instance, if we write A1, A2, . . . , A132, we know that there areexactly 132 sets. (Try writing them down using different letters of the alphabet!)

• We can even talk about an infinite number of sets A1, A2, A3, . . .. Notice thatthe subscripts all come from the set N. We refer to N as the index set for thiscollection.

• Using indices we can form complicated unions, intersections, and Cartesianproducts.

Example 2.7. Let A1 = {1, 2, 4}, A2 = {−3, 1, 5, 9}, and A3 = {1, 6, 10}. We find

3⋃

i=1

Ai = A1 ∪ A2 ∪ A3 = {−3, 1, 2, 4, 5, 6, 9, 10}

and3⋂

i=1

Ai = A1 ∩ A2 ∩ A3 = {1}.

Those who have seen summation notation3∑

i=1

i2 = 12 + 22 + 32

will recognize where this notation comes from. △Can we form infinite unions and intersections? This is actually a common occur-

rence.

Example 2.8. Let B1 = {1,−1}, B2 = {2,−2}, B3 = {3,−3}, and so forth. In otherwords Bn = {n,−n} for each n ∈ N. (Notice that while the subscripts come from N,the elements of the sets Bn come from Z.)

The union is the set of elements which belong to at least one of the sets, thus∞⋃

n=1

Bn = {. . . ,−3,−2,−1, 1, 2, 3, . . .} = Z− {0}.

The intersection is the set of elements which belong to every one of the sets, thus∞⋂

n=1

Bn = { } = ∅. △

There is an alternate way to write intersections and unions, using index sets. Forinstance, using the notation in the previous two examples, we could also write

3⋂

i=1

Ai = A1 ∩ A2 ∩ A3 =⋂

i∈{1,2,3}

Ai

and∞⋃

n=1

Bn =⋃

n∈N

Bn.

There is nothing to limit our index set, so we can make the following broad definition.


Definition 2.9. Let I be any set, and let Si be a set for each i ∈ I. We put

⋃

i∈I

Si = {x : x belongs to Si for some i ∈ I}

and

⋂

i∈I

Si = {x : x belongs to Si for each i ∈ I}.

The next example shows, once again, how mathematics has the uncanny abilityto express information in varied subjects using very simple notation.

Example 2.10. Let A = {a,b,c,d, . . . , z} be the “lowercase English alphabet set.”This set has twenty-six elements. Let V = {a,e,i,o,u} be the “standard vowel set.”Notice that V ( A.

Given α ∈ A, we let Wα be the set of words in the English language containingthe letter α. Note that α is a dummy variable, standing in for an actual element ofA. For instance, if α = x then we have

Wx = {xylophone, existence, axiom, . . .},while if α = t then we have

Wt = {terminator, atom, attribute, . . .}.Each set of words Wα is a subset of the universal set of all words in the Englishlanguage.

Try to answer the following questions:(1) What is

⋂α∈V Wα?

(2) Is that set empty?(3) What is

⋃α∈V Wα?

(4) Is that set empty?Here are the answers. (Look at them only after you have your own!)

(1) This is the set of words that contain every standard vowel.(2) It isn’t empty, since it contains words like “sequoia,” “evacuation,” etc.(3) This is the set of words with no standard vowels. (Don’t forget that there is a

bar over the union.)(4) It isn’t empty, since it contains words like “why,” “tsktsk,” etc. △We finish with one more difficult example.

Example 2.11. We determine⋃

x∈[1,2][x, 2]× [3, x+ 3].First, to get a footing on this problem, we try to understand what happens for

certain values of x. The smallest possible x value in the union is when x = 1.There we get that [x, 2] × [3, x + 3] = [1, 2] × [3, 4]. This is the set of ordered pairs{(x, y) ∈ R2 : 1 ≤ x ≤ 2, 3 ≤ y ≤ 4}. This is just a box in the plane. Its graph isthe first graph below.


x

y

[1, 2]× [3, 4]

x

y

{2} × [3, 5]

The largest possible x value in the union is when x = 2. There we get that[x, 2] × [3, x + 3] = [2, 2] × [3, 5]. Notice that [2, 2] = {2} is just a single point. Now{2} × [3, 5] is a line segment in the plane, where the x-value is 2 and the y-valuesrange from 3 to 5. Its graph is the second graph above.

If we consider the intermediate value x = 1.5, we get the box [1.5, 2] × [3, 4.5],graphed below on the left.

x

y

[1.5, 2]× [3, 4.5]

x

y

⋃

x∈[1,2]

([x, 2]× [3, x+ 3])

Taking the union over all x ∈ [1, 2], we get the region graphed above on the right,boxed in by the lines x = 1, y = 3, x = 2, and y = x+ 3.

△

2.C Exercises

Exercise 2.1. Sketch each of the following sets in the Cartesian plane R2.(a) {1, 2} × {1, 3}.(b) [1, 2]× [1, 3].(c) (1, 2]× [1, 3]. (Hint: If an edge is missing, use a dashed, rather than solid, line

for that edge.)(d) (1, 2]× {1, 3}.


Exercise 2.2. Let A = {s, t} and B = {0, 9, 7}. Write the following sets by listingall of their elements.(a) A× B.(b) B × A.(c) A2.(d) B2.(e) ∅ × A.

Exercise 2.3. Answer each of the following questions with “True” or “False” andthen provide a reason for your answer.(a) If |A| = 3 and |B| = 4, then |A× B| = 7.(b) It is always true that A×B = B × A when A and B are sets.(c) Assume I is an indexing set, and let Si be a set for each i ∈ I. We always have⋂

i∈I Si ⊆⋃

i∈I Si.(d) There exist distinct sets S1, S2, S3, . . ., each of which is infinite, but

∞⋂

i=1

Si

has exactly one element.(e) The set A4 consists of ordered triples from A.

Exercise 2.4. Using the notations from Example 2.10, write the following sets (pos-sibly using intersections or unions).(a) The set of words containing all four of the letters “a,w,x,y.”(b) The set of words not containing any of the letters “s,t,u.”(c) The set of words containing both “p,r” but not containing any of the standard

vowels. (Is this set empty?)

Exercise 2.5. For each number r ∈ R, consider the “parabola shifted by r” definedas:

Pr = {(x, y) ∈ R2 : y = x2 + r}.Describe the following sets in set-builder notation; the answer should have no referenceto “r.” Also graph the sets in the Cartesian plane.(a)

⋃r∈R Pr.

(b)⋃

r>0 Pr.(c)

⋃r 6=0 Pr.

(d)⋂

r∈R Pr.(e)

⋂r>0 Pr.


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