CSE 20 – Discrete Mathematics

1. Peer Instruction in Discrete Mathematics by Cynthia Leeis licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.Based on a work at http://peerinstruction4cs.org.Permissions beyond the scope of this license may be available at http://peerinstruction4cs.org. CSE 20 – Discrete Mathematics Dr. Cynthia Bailey Lee Dr. Shachar Lovett

2. Today’s Topics: • Set sizes • Set builder notation • Rapid-fire set-theory practice

3. 1. Set sizes

4. Power set • Let A be a set of n elements (|A|=n) • How large is P(A) (the power-set of A)? • n • 2n • n2 • 2n • None/other/more than one

5. Cartesian product • |A|=n, |B|=m • How large is A x B ? • n+m • nm • n2 • m2 • None/other/more than one

6. Union • |A|=n, |B|=m • How large is A  B ? • n+m • nm • n2 • m2 • None/other/more than one

7. Intersection • |A|=n, |B|=m • How large is A  B ? • n+m • nm • At most n • At most m • None/other/more than one

8. 2. Set builder notation

9. Set builder notation • Example: Even • Our definition of Even is: • How can we write this as a set, rather than a definition applying to an individual n? • , or just • “|” is pronounced “such that”

10. Set builder notation • How could we write “the set of integers that are multiples of 12”? • Other/none/more than one

11. Ways of defining a set • Enumeration: • {1,2,3,4,5,6,7,8,9} • + very clear • - impractical for large sets • Incomplete enumeration (ellipses): • {1,2,3,…,98,99,100} • + takes up less space, can work for large or infinite sets • - not always clear • {2 3 5 7 11 13 …} What does this mean? What is the next element? • Set builder: • { n | <some criteria>} • + can be used for large or infinite sets, clearly sets forth rules for membership

12. Primes • Enumeration may not be clear: • {2 3 5 7 11 13 …} • How can we write the set Primes using set builder notation?

13. 3. Rapid-fire set-theory practice Clickers ready!

14. Set Theory rapid-fire practice • (A and B are sets) • TRUE • FALSE In your discussion: If true, prove it! (quickly sketch out what the argument would be) If false, what are the counterexample A and B?

15. Set Theory rapid-fire practice • (A and B are sets) • TRUE • FALSE In your discussion: If true, prove it! (quickly sketch out what the argument would be) If false, what are the counterexample A and B?

16. Set Theory rapid-fire practice • (A is a set) • TRUE • FALSE In your discussion: If true, prove it! (quickly sketch out what the argument would be) If false, what are the counterexample A and B?