Comparing sets

1. Comparing sets Chapter 2 Sec 2

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. • Def • Two 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.

3. 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}

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

5. 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

6. 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.

7. 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,

8. 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

9. Example • , which is true. • Also because {1,2,3,…} contains elements that are not members of {2,4,6,…}.

10. Exercise • Find all the subsets of {1,2,3} • If a set has five elements, how many subsets will it have? • 25