Sets and relations

Reading: Chapter 5 (72-93) from the text book

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Sets

• We’ll look briefly at the main ideas of sets• Our intention is to introduce terminology &

notation that will be useful later• The term set means a collection of items• The items are called the elements of the set• A set can be described in 2 ways –

1. in enumerated form (i.e. as a list)2. in predicate form (i.e. using a property that

defines the elements of the set)2

Examples of Enumerated Sets

• the set of summer months is {June, July, August} note the use of braces (‘curly brackets’) for sets

• the set of positive even numbers less than 10 is {2, 4, 6, 8}

• the set of positive even numbers less than 100 is {2, 4, 6, 8, …, 98} – an ellipsis (3 dots) is used if there is a clear pattern

• the set of positive even numbers is {2, 4, 6, 8, …}

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Examples of Sets in predicate form

• The set of positive even numbers less than 100 can be written in predicate form as {x: x is even and 0 < x < 100}• This definition is read as ‘the set of all x such that x is even and 0 < x < 100’• Sets are usually denoted by capital letters e.g. A = {2, 4, 6, 8}, B = {x: x is a pos. even no.}• The symbols ∈ and ∉ mean ‘is an element of’ and ‘is not an element of’, respectively• e.g. 6 ∈ A, 120 ∈ B, 7 ∉ A

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Symbols for Special Sets

• Special symbols are used for certain sets of nos:N = set of natural numbers = {1, 2, 3, 4, …}J = set of integers = {…, –3, –2, –1, 0, 1, 2, 3,…}Q = set of rational nos = {x: x = m/n for some integers m and n with n ≠ 0}R = set of all real nos

∅ = the null set or empty set. It has no elements, & may be written as { } or even as {x: x = x + 1} (any predicate that is always false can be used)

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The Universal Set

• The universal set, denoted by U, contains all elements that could be under discussion in a particular situation

• U changes according to circumstances• e.g. If we’re dealing with months of the year,

U = {January, February, March, …, December} If we’re dealing with numbers, U might be R (the set of all real nos)

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Subsets, Set Operations andVenn Diagrams

• If A& B are sets so that every element of B is an element of A, B is a subset of A (written B ⊆ A)

• e.g. A = {1,2,3,4}, B = {1,3,4}, C = {4,5,6}. Then B ⊆ A, but C is not a subset of A.

• In a picture:

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Venn Diagrams

• A picture such as in the previous slide is called a Venn diagram

• Venn diagrams were introduced by John Venn, who used them in his book Symbolic Logic (1881) to illustrate principles of logic

• Venn diagrams are easy to use for 2 or 3 sets.• For more than 3 sets, the diagrams become

quite complicated and are not so easy to use.

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Properties of Sets

• Recall: If every element of B is an element of A, B is a subset of A, written as B ⊆ A• Thus, for any set A, it is true that A ⊆ A• Also, for any set A, it is true that ∅⊆ A i.e.we can’t find an element of ∅ which isn’t in A• Two sets A and B are equal if A ⊆ B and B ⊆ A

Thus 2 sets are equal if they have the same elts• So the order of listing elts is immaterial

e.g. {1, 2, 3} = {2, 1, 3} – & there’s no reason to list an elt more than once – e.g. {1, 2, 1} = {1, 2}

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Set Operations• The intersection of 2 sets A and B is A ∩ B = {x: x ∈ A and x ∈ B}• 2 sets A and B are disjoint if A ∩ B = ∅ (i.e. if the

sets have no elements in common)• The union of 2 sets A and B is A ∪ B = {x: x ∈ A or x ∈ B}

where ‘or’ means the inclusive ‘or’ • The complement of a set A consists of all the

elements of the universal set that are not in A. i.e.

= {x : x ∈U and x ∉ A}A

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Set Operations

• The difference of 2 sets A and B is A – B = {x: x ∈ A and x ∉ B}

A – B isshaded in red

• Note that A − B = A ∩• This can be shown using the defns of set operations, or by using Venn diagrams

B

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Example

Suppose E = {a, b, c, d, e, f, g, h, i, j},A = {a, b, c, d, e, f, g}, B = {b, d, f, i, j},C = {a, c, f, j}. Find:

(i) A ∪ C(ii) A ∩ B(iii) A∩ C(iv) (B ∩ A) ∪ C(v) A (∪ C ∩ B)

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Laws of Sets

Laws of Sets Name

AA

absorption

inverse

onannihilati

identitysMorgan’ de

vedistributi

eassociativ

ecommutativ

idempotent

complement double

A B) (A A

A A

A

A A

B A B AC) (A B) (A

C) (B A C) (B A

C B) (A

A B B A

A A A

U

UU

A B) (A A

A A

A

A UA

B A B AC) (A B) (A

C) (B A C) (B A

C B) (A

A B B A

A A A

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Verifying and Using the Laws of Sets

• All the laws of sets can be verified• Another way of verifying the laws is to use Venn

diagrams• Example: Use Venn diagrams to illustrate the 2nd

de Morgan’s law for sets• The laws of sets can be used to simplify a given

set (just as we will use the laws of Algebra to simplify a given algebraic expression)

• Example: Use laws of sets to simplify A)BA( 14

The Power Set

• Suppose A = {a, b}. The subsets of A are ∅, {a}, {b} & {a, b}

• The set of these subsets is called the power set of A, denoted by P (A) i.e.

P(A) = { , {∅ a}, {b}, {a, b}}• Note that P (A) is a set whose elements are

themselves sets – i.e. it is a set of sets• Also note that A has 2 elements, & P (A) has 4

elements• Exercise: If A = {a, b, c}, write down P(A) 15

Cardinality of the Power Set• The cardinality of a finite set A is the no. of

elements in the set, written as | A |• Example: If A = {a, b, c}, then | A | = 3• Observe that A has 3 elements, & P(A) has 8

elements• This leads to the general observation: If A has n elements, then P(A) has 2n elements i.e. if | A | = n, then |P(A) | = 2n

• Then a set with n elements has 2n subsets16

Ordered Pairs

• When dealing with sets, the ordering of elements inthe set is immaterial – e.g. {2, 1, 4} = {1, 4, 2}

• Sometimes, though, order does matter e.g.: (i) a list of place-getters in a race, or a list offootball teams in order of leader position;

(ii) an ordered string of characters such as atax file no., password, credit card PIN or car reg. no.

• An ordered pair is a pair of elements in a particularorder, written as (a, b)

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Ordered n-tuples• Thus the ordered pair (3, 5) is different to (5, 3)• Note the use of parentheses (‘round brackets’),

and not braces (‘curly brackets’) as for sets• If we have n elements, an ordered n-tuple is a

list of the n elements in a particular order – it iswritten as (x1, x2, x3,…, xn)

• Since order is important, the only way for (x1, x2, x3,…, xn) = (y1, y2, y3,…, yn) is if the 1st

elements are the same (i.e. x1 = y1), the 2nd

elements are the same (i.e. x2 = y2), and so on• So (1, 4, 5) ≠ (1, 5, 4) (but {1, 4, 5} = {1, 5, 4})

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The Cartesian Product of 2 Sets• The Cartesian product of 2 sets A and B is A × B = {(x, y): x ∈ A and y ∈ B}

i.e. It is the set of all ordered pairs, where thefirst element is from A & the second elementis from B

• e.g. If A is the set of digits 0-9, & B is the set ofletters a-z, then (3, t) is in A × B, but (m, 7) is notin A × B – although it is in B × A

• e.g. If A = {1, 2, 3} & B = {p, q}, then A × B = {(x, y): x ∈ A and y ∈ B} = {(1, p), (1, q), (2, p), (2, q), (3, p), (3, q)} 19

The Cartesian Product of n Sets

• The Cartesian product of n sets A1, A2,…, An is A1 × A2 ×… × An =

{(x1, x2, x3,…, xn): x1 ∈ A1, x2 ∈ A2, …, xn ∈ An}• i.e. It is the set of all ordered n-tuples, where

the 1st elt is from A1, the 2nd elt is from A2,etc• e.g. A car reg. no. such as KCT454 can be

regarded as an ordered 6-tuple (K, C, T, 4, 5,4).• If L is the set of all letters, & D is the set of all

decimal digits, then the set of all possible carregistration nos is L × L × L × D × D × D 20

Cartesian Product of a Set with Itself

• The set A × A ×… × A (n times) is written as An

• e.g. If R is the set of real nos, then R2 is the set of all ordered pairs (x, y), where x& y are real nos – geometrically, R2 is the 2-dimensional plane

• Similarly, think of R3 as all points in 3-dim space

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Cartesian Product of a Set with Itself

• e.g. {0, 1}2 = {(0, 0), (0, 1), (1, 0), (1, 1)}

• e.g. The elements of {0, 1}n are ordered n-tuplesin which each element is 0 or 1 – so a typicalelement of {0, 1}6 is (0, 1, 1, 1, 0, 1)

• Think of {0, 1}n as the set of all strings of n bits

• Note that L × L × L × D × D × D = L3 × D3

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Computer Repn of Sets• To enable computers to handle sets, assume

the elements of the universal set U are listed in a definite order.

• Then, if |U| = n and A is a set, A is representedby a string of n bits b1b2b3…bn.

• Here bi is 1 if the ith elt of U is in A, and bi is 0if the ith elt of U is not in A.

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Computer Repn of Sets

• Example: Suppose U = {a, b, c, d, e, f, g}. Find: (a) the representation of {d, f, a, g} as a bit string (b) the set represented by the bit string 0111011

• For sets defined w.r.t. the same universal set, theoperations of intersection, union & complementcan be carried out directly on the bit strings, without having to convert to the original sets.

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Computer Repn of Sets• The bit string of A ∩ B has a 1 if the bit strings of

A & B both have a 1, & otherwise has a 0

• This process is termed a bitwise and operation

• The bit string of A ∪ B has a 0 if the bit strings of A & B both have a 0, & otherwise has a 1 (this is a bitwise or operation)

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Set Operations Using Bit Strings

• The bit string for the complement of A isobtained from that of A by simply replacing 0with 1, and 1 with 0 (a bitwise not operation)

• Example: Suppose the bit strings of A& B are A: 0110110101, B: 1111001001 Find the bitstrings of A ∩ B, A ∪ B &

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A

Relations• A binary relation occurs when we say

something about a property of an object relative to another object of the same type

• Example: The statement ‘Ali is taller than Yasir’ illustrates a relation

• The word ‘binary’ refers to the fact that two objects are compared – in future, we’ll omit this word and refer to just a ‘relation’ 27

Examples which Illustrate Relations

• Examples of statements from everyday life which illustrate relations:

‘Ali is the husband of Alia’‘Nadir is the sister of Nuha’‘Australia has a smaller population than

China’‘Discrete Maths is a prerequisite for

Encryption and Network Security’28

More Examples which Illustrate Relations

• Examples of statements from mathematics which illustrate relations:

‘12 is greater than 4’‘{a} is a subset of {a, b, c}’‘20 is divisible by 4’‘Line L1 is parallel to line L2’

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Comments on the Examples• In each of the examples, a statement is made

about a pair of objects of the same type.

• The order of the objects is often important – e.g. it is true that ‘Australia has a smaller population than China’, but it is not true that ‘China has a smaller population than Australia’

• Thus relations involve 2 objects of the same type (i.e. from the same set), where order is important30

Definitions of a Relation• Informal Defn: A relation can be thought of as

a statement about ordered pairs (x, y) that are in A × A, where A is some set.

• This is the basic idea of a relation, although the formal definition looks a little different.

• Formal Defn: A (binary) relation on a set A is a subset R of A × A. We say that x & y are related iff (x, y) ∈ R. 31

Example• Consider the relation ‘is greater than’ on the

set A = {3, 5, 6, 8}.

• For any (x, y) ∈ A × A, either x is greater than y, and then x is related to y or x is not greater than y, and then x is not related to y.

• The set of the ordered pairs (x, y) ∈ A × A, for which x is related to y is given by: R = {(5, 3), (6, 3), (8, 3), (6, 3), (6, 5), (8, 6)} 32

Notation for a Relation

• In the previous example, we can state that x & y are related by writing (x, y) ∈ R.

• In practice, this is often written as xRy (read this as ‘x is related to y’).

• For the previous example, we can write ‘x > y’ instead of xRy to mean that x is related to y

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Graphical Repn of a Relation

• Example: The reln ‘>’ on the set A = {3, 5, 6, 8} can be depicted using a graph. The elements of A are represented by dots, & if x is related to y, an arrow is drawn from x to y. The result is called a directed graph.

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Matrix Repn of a Relation

• A relation can also be represented by a matrix (plural ‘matrices’) called the relation matrix.

• The entry in row x & column y is T if x is related to y, and is F otherwise.

• e.g. For ‘>’ on {3, 5, 6, 8}, the relation matrix is given by

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