1. CSE 20 – Discrete Mathematics Dr. Cynthia Bailey Lee Dr. Shachar Lovett 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.Cynthia LeeCreative Commons Attribution- NonCommercial-ShareAlike 4.0 International Licensehttp://peerinstruction4cs.org

2. Today’s Topics: 1. Set Theory: Venn Diagrams 2. Set Theory: Proving two sets are equal 3. Set Theory: Power Set and Cross Product 2

3. Set notations  Union: A  B = {x:x  A  x  B}  Intersection: A  B = {x:x  A  x  B}  Complement:A’=A c = {x:x  A}  Difference:A-B = {x:x  A  x  B}  Empty set:  3

4. 1. Venn Diagrams 4

5.  We will prove this using Venn Diagram method  First, we need to draw a generic Venn Diagram: 5 U B A C

6. 6 U B A C

7. 7 U B A C

8. 8 U B A C

9. 9

10. 2. Proving two sets are equal 10

11. Set equality 11

12. Set equality  Another take: one way to prove that X=Y is to prove that both  X  Y  Y  X 12

13. Set equality  Can also prove directly  Claim: (A’)’=A  Proof: A’={x: x  A} (A’)’ = {x: x  A’} = {x: x  A} 13

14. 3. Power Set and Cross Product 14

15. Power Sets 15

16. Cross Product 16