SetsLesson 2.1

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SetA set is an unordered collection of well-defined objects.

Example:The set V of vowels in the English alphabet can be written as V ={a, e, i, o, u} .

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Elements of a SetThe objects in a set are called the elements of the set. A set is said to contain its elements.

Example:If V ={a, e, i, o, u}, then we can say a is an element of V (we write a V). The other elements of V are e, i, o and u.

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Set-builder NotationAnother way to describe a set is to use the set-builder notation.

Example:V ={x | x is a vowel in the English alphabet}is read as “V is the set of all x’s such that x is vowel in the English alphabet”.

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Equal SetsTwo sets A and B are equal (A=B) if and only if they have the same elements. That is,

A=B if and only if .( )x x A x B

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Example of Equal Sets If A={1, 3, 5} and B={3, 5, 1}, then A=B

because they have same elements.

If A={1, 3, 5} and B={3, 5, 1, 6}, then A is not equal to B because 6 is an element of Bbut 6 is not an element of A. We write A B.

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Venn DiagramSets can be represented graphically using Venn diagrams. In Venn diagrams, the universal set U, which contains all the objects under consideration, must be defined.

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Example of Venn DiagramLet the universal set U be set of all letters in the English alphabet. Let V = { a, e, i, o, u}.The the Venn Diagram of the set V is:

UV

a e i o u

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SubsetThe set A is said to be a subset of B (denoted by A B) if and only if every element of A is also an element of B. That is,

A B if and only if

( )x x A x B

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Example of Subset If A={1, 3, 5} and B={3, 5, 1, 4}, then A B

because all elements of A are also elements of B.

However, B is a not subset A since 4 is an element of B but 4 is not an element of A. We write B A.

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Empty Set The special set with no elements is called the

empty set or null set.

We denote the empty set by or { }.

Note that

{ }

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Theorem on Empty SetTheorem 1. For every set S, . Proof:

Let S be a set. To show , we must showis always true. Note that

is always false since has no element. Because (F→T) and (F→ F) are always true statements, we conclude

is always true.

S

S ( )x x S x

( )x x S

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Cardinality of a SetThe number of elements of a set S is called the cardinality of S. We denote the cardinality of S by | S |.

Example:If V ={a, e, i, o, u}, then |V| =5.

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Finite SetIf the number of elements of a set is a definite integer, then it is called a finite set.

Example:Let A = {a, b, c, d}. Since |A| = 4, A is a finite set.

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Infinite SetA set is said to be infinite if it is not finite.

Example:Consider the set of natural numbersN = {1, 2, 3, 4, 5, 6, …}. Since number of

elements of N can never be a definite integer, N is not finite. Therefore, N is infinite.

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Power SetGiven a set S, the power set is the set of all subsets of S. We denote it by P(S).Note: If S has n elements, then P(S) has 2n

elements. Example: Let S = {1, 2, 3}. Then P(S)={ ,{1}, {2}, {3}, {1,2},

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

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Cartesian Product of SetsLet A and B be sets. The Cartesian Product of A and B, denoted by A B, is defined by

A B = {(a,b) | a A and b B}Example:Let A={a, b, c} and B={1, 2}.Then A B = {(a,1), (a,2), (b,1), (b,2),

(c,1), (c,2)}.