COMPARING SETS

Chapter 2 Sec 2

SET EQUALITY One of the fundamental things

we need to know about two sets is when do we consider them to be the same.

DefTwo sets A and B are equal if they have exactly the same members. In this case, we write A = B. If A and B are not equal, we write A ≠ B.

EXAMPLE, ARE THE SETS EQUAL? {Socrates, Shakespeare, Armstrong} = {Armstrong, Socrates, Shakespeare}

A={x:x is a citizen of the US} and B={y:y was born in the US}

SUBSETS

Another way we compare sets is to determine whether one set is part of another set.

DEFINITION The set A is a subset of the set B if every element of A is also an element of B. We indicate this relationship by writing . If A is not a subset of B, then we write

BA

BA

In order to show that , we must show that every element of A also occurs as an element of B. To show that A is not a subset of B, all we have to do is find one element of A that is not in B.

BA

IDENTIFYING SUBSETS Determine whether either set is a subset of the other.

A ={2, 5, 6} and B ={1,2, 5, 6}Every member of A is in B, therefore we can write .

But, there is an element of B that is not in A,

BA

AB

PROPER SUBSET The set A is a proper subset of the set B if but A ≠ B.

We write this as . If A is not a proper subset of B, then we write

BA

BA

BA

EXAMPLE , which is true.

Also because {1,2,3,…} contains elements that are not members of {2,4,6,…}.

,...}3,2,1{,...}8,6,4,2{

,...}3,2,1{,...}8,6,4,2{

EXERCISE Find all the subsets of {1,2,3}

If a set has five elements, how many subsets will it have?25