1. CSE 20 DISCRETE MATH Prof. Shachar Lovett http://cseweb.ucsd.edu/classes/wi15/cse20-a/ Clicker frequency: CA

2. Todays topics Equivalence relations Section 6.2 in Jenkyns, Stephenson

3. (Binary) Relations

4. Three important properties of relations

5. Examples of equivalence relations U=Z (integers) xRy = “x=y” Prove that R is an equivalence relation

6. Examples of equivalence relations U=Z (integers) xRy = “|x|=|y|” (absolute value of x = absolute value of y) Prove that R is an equivalence relation

7. Examples of equivalence relations U=“all students in this class” xRy = “x and y have the same birthday” Prove that R is an equivalence relation

8. Examples of equivalence relations U=Z (integers) xRy = “x+y is even” Prove that R is an equivalence relation

9. What is so special about equivalence relations? Equivalence relations describe a partition of the universe U to equivalence classes

10. Equivalence classes

11. Equivalence classes: example 1 U=Z (integers) xRy = “x=y” Class(x)= A. x B. y C. U D. “x=y” E. Other

12. Equivalence classes: example 2 U=Z (integers) xRy = “|x|=|y|” Class(x)= A. x B. |x| C. “x=y” D. “|x|=|y|” E. Other

13. Equivalence classes: example 3 U=“all students in this class” xRy = “x and y have the same birthday” Class(x)= A. x B. Birthday of x C. “x=y” D. “|x|=|y|” E. Other

14. Equivalence classes: example 4 U=Z (integers) xRy = “x+y is even” Class(x)=??? Figure this out in your groups

15. Equivalence classes: proof

16. Equivalence classes: proof (contd)

18. Next class Modular arithmetic Section 6.2 in Jenkyns, Stephenson Review session for midterm 2