1. MAT 2720 Discrete Mathematics Section 3.3 Relations http://myhome.spu.edu/lauw

2. Goals Relations Properties of Relations on X

3. Recall A relation from X to Y is a subset Sometimes, we write Domain of R = all possible value of x Range of R = all possible value of y

4. Recall A relation from X to X is called a relation on X

5. Properties of Relation on X R is…If…Diagraph Reflexive Symmetric Transitive

6. Example 5(a)

7. Example 5(b)

8. Example 5(c)

9. Properties of Relation on X R is…If…Diagraph Antisymmetric(Read)

10. MAT 2720 Discrete Mathematics Section 3.4 Equivalence Relations http://myhome.spu.edu/lauw

11. Goals Equivalence Relations A special relation with nice properties. Partition of sets (Clumping Property). Applications to counting problems. CS students should read the applications in p.166-168

12. “Informal” Example Example

13. “Informal” Example Reflexive?

14. “Informal” Example Symmetric?

15. “Informal” Example Transitive?

16. “Informal” Example “Clumping” Effective

17. Definitions and Notations is an Equivalence Relation if R is reflexive, symmetric, and transitive.

18. Example 1 Show that R is an Equivalence Relation

19. Example 1 Show that R is an Equivalence Relation

20. Example 1 Proof: ReflexiveAnalysis

21. Example 1 Proof: SymmetricAnalysis

22. Example 1 Proof: TransitiveAnalysis

23. Definitions and Notations is an Equivalence Relation if R is reflexive, symmetric, and transitive. Equivalence Class of :

24. Example 1

25. Observations

27. Partition of a Set (1.1) A partition of a set X is a way to split X into the union of disjoint subsets.

28. Partition of a Set (1.1) A partition of a set X is a way to split X into the union of disjoint subsets. For every element in X, it belongs to one and only one subset in the partition.

29. Theorem

30. “Informal” Example Partition

31. Theorem

32. (It is easy to check that R is an equivalence relation.) Example 2

33. Summary of the 2 Theorems

34. Theorem