Sets

1

Sets

Informally:

A set is a collection of (mathematical) objects,

with the collection treated as a single mathematical object.

• real numbers,

• complex numbers, C

• integers,

• All students in our class

Examples:

Defining Sets

Sets can be defined directly:

e.g. {1,2,4,8,16,32,…},

{CSC1130,CSC2110,…}

Order, number of occurence are not important.

e.g. {A,B,C} = {C,B,A} = {A,A,B,C,B}

A set can be an element of another set.

{1,{2},{3,{4}}}

Defining Sets by Predicates

The set of prime numbers:

){ }(|x A P x

The set of elements, x, in A such that P(x) is true.

Commonly Used Sets

• N = {0, 1, 2, 3, …}, the set of natural numbers• Z = {…, -2, -1, 0, 1, 2, …}, the set of integers• Z+ = {1, 2, 3, …}, the set of positive integers• Q = {p/q | p Z, q Z, and q ≠ 0}, the set of rational

numbers• R, the set of real numbers

Special Sets

• Empty Set (null set): a set that has no elements, denoted by ф or {}.

• Example: The set of all positive integers that are greater than their squares is an empty set.

• Singleton set: a set with one element • Compare: ф and {ф}

– Ф: an empty set. Think of this as an empty folder– {ф}: a set with one element. The element is an empty set.

Think of this as an folder with an empty folder in it.

Venn Diagrams

• Represent sets graphically• The universal set U, which contains all the objects under

consideration, is represented by a rectangle. The set varies depending on which objects are of interest.

• Inside the rectangle, circles or other geometrical figures are used to represent sets.

• Sometimes points are used to represent the particular elements of the set.

7

a

V eio

uU

{7, “Albert”, /2, T}

Membership

x A x is an element of A

x is in A

/2 {7, “Albert”,/2, T}

/3 {7, “Albert”,/2, T}

14/2 {7, “Albert”,/2, T}

7 2/3

Examples:

Containment

A B A is a subset of B

A is contained in B

Every element of A is also an element of B.

Examples: R, {3}{5,7,3}

every set, A A

A is a proper subset of B

Set Equivalence

Two sets are equal if and only if they have the same elements. That is, if A and B are sets, then A and B are equal if and only if

We write A = B if A and B are equal sets.

B). xA x(x

• Example:• Are sets {1, 3, 5} and {3, 5,1} equal?• Are sets {1, 3, 3, 3, 5, 5, 5, 5} and {1, 3, 5}

equal?

Basic Operations on Sets

:: { | ( ) ( )}A B x x A x B union:

:: { | }A B x x A x B intersection:

Basic Operations on Sets

Basic Operations on Sets

difference: :: { | ( ) ( )}A B x x A x B

Basic Operations on Sets

:: { | } x D x A DA Acomplement:

Cardinality

• Example:– Let A be the set of odd positive integers less than 10.

Then |A| = 5.– Let S be the set of letters in the English alphabet. Then |

A| = 26.– Null set has no elements, | ф | = 0.

15

Let S be a set. If there are exactly n distinct elements in S where n is a

nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|.

Infinite Sets

• Example: The set of positive integers is infinite.

16

A set is said to be infinite if it is not finite.

Cardinality

• Finding the cardinality of |A U B|:|A U B| = |A| + |B| - |A ∩ B |

• Example: A = {1,3,5,7,9}, B = {5,7,9,11} |A U B| = |A| + |B| - |A ∩ B |

= 5 + 4 – 3 = 6

17

Partitions of Sets

Two sets are disjoint if their intersection is empty.

A collection of nonempty sets {A1, A2, …, An} is a partition of a set A

if and only if

A1, A2, …, An are mutually disjoint.

Power Sets

( ) :: { |o }p w A S S A

pow( , ) , , , ,a b a b a b

power set:

Cartesian Products

• Sets are unordered, a different structure is needed to represent an ordered collections – ordered n-tuples.

• Two ordered n-tuples are equal if and only if each corresponding pair of their elements is equal.– (a1, a2,…, an) = (b1, b2,…, bn) if and only if ai = bi for i = 1, 2,

…, n

20

The ordered n-tuple (a1, a2,…, an) is the ordered collection that has a1

as its first element, a2 as its second element, …, and an as its nth element.

Cartesian Products

• Example: What is the Cartesian product of A = {1,2} and B = {a,b,c}?Solution:

A × B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}• Cartesian product of A × B and B × A are not equal, unless A =

ф or B = ф (so that A × B = ф ) or A = B. B × A = {(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)}

21

Let A and B be sets. The Cartesian product of A and B, denoted by

A × B, is the set of all ordered pairs (a, b), where a A and b B.

Hence,A × B = {(a,b)| a A Λ b B}.

Cartesian products

• Example: What is the Cartesian product of A × B × C where A= {0,1}, B = {1,2}, and C = {0,1,2}?Solution:

A × B × C= {(0,1,0), (0,1,1), (0,1,2), (0,2,0), (0,2,1), (0,2,2), (1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)}

22

The Cartesian product of sets A1, A2, …, An, denoted by A1 × A2

× … × An is the set of ordered n-tuples (a1, a2, …, an), where ai

belongs to Ai for i = 1,2, …, n. In other words,

A1 × A2 × … × An = {(a1, a2, …, an) | ai Ai for i = 1,2, …, n}.

Set Identities

Distributive Law:

Set Identities

De Morgan’s Law:

Proving Set Identities

Proving Set Identities

Computer Representation of Sets

• Represent a subset A of U with the bit string of length n, where the ith bit in the string is 1 if ai belongs to A and is 0 if ai does not belong to A.

• Example: – Let U = {1,2,3,4,5,6,7,8,9,10}, and the ordering of elements of U has

the elements in increasing order; that is ai = i. What bit string represents the subset of all odd integers in U? Solution: 10 1010 1010

What bit string represents the subset of all even integers in U?Solution: 01 010 10101What bit string represents the subset of all integers not exceeding 5 in U?

Solution: 11 1110 0000What bit string represents the complement of the set {1,3,5,7,9}?Solution: 01 0101 0101

27

Logic and Bit Operations

• Computers represent information using bits.• A bit is a symbol with two possible values, 0 and 1.• By convention, 1 represents T (true) and 0 represents F

(false).• A variable is called a Boolean variable if its value is

either true or false.• Bit operation – replace true by 1 and false by 0 in logical

operations.

Table for the Bit Operators OR, AND, and XOR.

x y x ν y x Λ y x y

0

0

1

1

0

1

0

1

0

1

1

1

0

0

0

1

0

1

1

0

28

Logic and Bit Operations

• Example: Find the bitwise OR, bitwise AND, and bitwise XOR of the bit string 01 1011 0110 and 11 0001 1101.

DEFINITION 7

A bit string is a sequence of zero or more bits. The length of this string is the number of bits in the string.

Solution:

01 1011 0110

11 0001 1101 ------------------- 11 1011 1111 bitwise OR

01 0001 0100 bitwise AND10 1010 1011 bitwise XOR

29

Set Operations

• The bit string for the union is the bitwise OR of the bit string for the two sets. The bit string for the intersection is the bitwise AND of the bit strings for the two sets.

• Example:– The bit strings for the sets {1,2,3,4,5} and {1,3,5,7,9} are 11 1110

0000 and 10 1010 1010, respectively. Use bit strings to find the union and intersection of these sets. Solution:Union:

11 1110 0000 V 10 1010 1010 = 11 1110 1010, {1,2,3,4,5,7,9}Intersection:

11 1110 0000 Λ 10 1010 1010 = 10 1010 0000, {1,3,5}

30

Let :: Sets |W S S S

S W S Sso

Russell’s Paradox

There is a male barber who shaves all those men, and only those men,

who do not shave themselves.

Does the barber shave himself?