SETS, FUNCTIONS AND RELATIONS, Slides of Discrete Mathematics

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Set Theory

S={1,2,4,5,3} 1 Є S 2 Є S S= {1,2,3}= {1,2,} S={1,2,3,…}

Set Builder Notation

S={1,2,3,…,100}

S= {x|x is a positive integer less than 100}

S= {x|xЄ Z+^ ^ x<100}

S= {x| p(x)}

Interval Notation

[a,b] closed interval include

(a,b) open interval exclude

1 < x < 10 = {2,3,4,5,…,9}

1<=x<=10 = {1,2,3…, 10}

[a,b)

(a,b]

Universal Set and Empty Set

The universal set U is the set containing

everything currently under

consideration.



Content depends on the context.



Sometimes explicitly stated, sometimes

implicit.

The empty set is the set with no

elements. Symbolized by ∅ or {}.

S={1,2,3,4,5,…,10}

A= {1,2}, B={3,5}, C={1,2,3},

{1,2,3,…10} A=B A B, B, A A= {1,2,3,5,5} B={1,3,2,5}

A={1,2,3,4,4,4,4}= {1,2,3,4},

|A|= 4

A={1,2,3,4}

B= {2,3,4,5,6}

AUB = {1,2,3,4, 2,3,4,5,6}= {1,2,3,4,5,6}= {1,4,3,5,6,2}

Fall 2002 CMSC 203 - Discrete Structures 20

Relations

If we want to describe a relationship between

elements of two sets A and B, we can use

ordered pairs with their first element taken

from A and their second element taken from B.

Since this is a relation between two sets , it is

called a binary relation.

Definition: Let A and B be sets. A binary relation

from A to B is a subset of AB.

In other words, for a binary relation R we have

R  AB. We use the notation aRb to denote that

(a, b)R and aRb to denote that (a, b)R.