1. Equivalence Relations Lecture 45 Section 10.3 Fri, Apr 8, 2005

2. Equivalence Relations An equivalence relation on a set A is a relation on A that is reflexive, symmetric, and transitive. We often use the symbol ~ as a generic symbol for an equivalence relation.

3. Examples of Equivalence Relations Which of the following are equivalence relations? a  b, on Z +. gcd(a, b) > 1, on Z +. A  B, on  (U). p  q, on a set of statements. p  q, on a set of statements. a  b (mod 10), on Z.

4. Examples of Equivalence Relations Which of the following are equivalence relations? p  q = p, on a set of statements. gcd(a, b) = 1, on Z +. gcd(a, b) = a, on Z +. A  B = , on  (U).  A  =  B , on  (U).

5. Examples of Equivalence Relations Which of the following are equivalence relations? R  R, on R. , on R.

6. Equivalence Classes Let ~ be an equivalence relation on a set A and let a  A. The equivalence class of a is [a] = {x  A  x ~ a}.

7. Examples: Equivalence Classes Describe the equivalence classes of each of the following equivalence relations. a  b (mod 10), on Z.  A  =  B , on  (U). p  q, on a set of statements. R  R, on R.

8. Equivalence Classes and Partitions Theorem: Let ~ be an equivalence relation on a set A. The equivalence classes of ~ form a partition of A. Proof: We must show that The equivalence classes are pairwise disjoint, The union of the equivalence classes equals A.

9. Equivalence Classes and Partitions Proof that the equivalence classes are pairwise disjoint. Let [a] and [b] be two distinct equivalence classes. Suppose [a]  [b]  . Let x  [a]  [b]. Then x ~ a and x ~ b. Therefore, a ~ x and x ~ b.

10. Equivalence Classes and Partitions By transitivity, a ~ b. Now let y  [a]. Then y ~ a. By transitivity, y ~ b. So y  [b]. Therefore, [a]  [b]. By a similar argument, [b]  [a].

11. Equivalence Classes and Partitions Thus, [a] = [b], which is a contradiction Therefore, [a]  [b] = . Thus, the equivalence classes are pairwise disjoint.

12. Equivalence Classes and Partitions Proof that the union of the equivalence classes is A. Let a  A. Then a  [a] since a ~ a. Therefore, a is in the union of the equivalence classes. So, A is a subset of the union of the equivalence classes.

13. Equivalence Classes and Partitions On the other hand, every equivalence class is a subset of A. Therefore, the union of the equivalence classes is a subset of A. Therefore, the union of the equivalence classes equals A. Therefore, the equivalence classes form a partition of A.

14. Example Let F be the set of all functions f : R  R. For f, g  F, define f ~ g to mean that f is  (g).

15. Example Theorem: ~ is an equivalence relation on F. Proof: Reflexivity Obviously, f ~ f for all f  F.

16. Example Symmetry Suppose that f ~ g for some f, g  F. Then f(x) is  (g(x)). There exist positive constants M 1, M 2, and x 0 such that M 1  g(x)    f(x)   M 2  g(x) , for all x > x 0.

17. Example It follows that (1/M 2 )  f(x)    g(x)   (1/M 1 )  f(x) , for all x > x 0. Therefore, g(x) is  (f(x)).

18. Example Transitivity Let f, g, h  F and suppose that f ~ g and g ~ h. Then there exist constants M 1 and x 1 and M 2 and x 2 such that  f(x)  M 1  g(x)  for all x  x 1 and

19. Example  g(x)  M 2  h(x)  for all x  x 2. Let x 0 = max(x 1, x 2 ). Then for all x  x 0,  f(x)  M 1  g(x)   M 1  M 2  h(x)  Therefore, f(x) is O(h(x)).

20. Example Similarly, we can show that h(x) is O(f(x)). Therefore, f(x) is  (h(x)). Therefore, f ~ h. Therefore, ~ is an equivalence relation on F.

21. Example The equivalence class of f is the set [f] of all functions with the same growth rate as f. The most important equivalence classes are [x a ], a  R, a > 0. [b x ], b  R, b > 1. [x a log b x], a  R, a > 0, b > 1.

22. Example Furthermore, [x a ]  [x b ] if a  b. [a x ]  [b x ] if a  b. However, [log a x] = [log b x] for all a, b > 1.

23. The Equivalence Relation Induced by a Partition Let A be a set and let {A i } i  I be a partition of A. Define a relation ~ on A as x ~ y  x, y  A i for some i  I.

24. The Equivalence Relation Induced by a Partition Theorem: The relation ~ defined above is an equivalence relation on A.

25. The Equivalence Relation Induced by a Partition Proof: We must prove that ~ is reflexive, symmetric, and transitive. Proof that ~ is reflexive. Let a  A. Then a is in A i for some i  I. So a ~ a.

26. The Equivalence Relation Induced by a Partition Proof that ~ is symmetric. Let a, b  A and suppose that a ~ b. Then a, b  A i for some i  I. So b, a  A i for some i  I. Therefore b ~ a.

27. The Equivalence Relation Induced by a Partition Proof that ~ is transitive. Let a, b, c  A and suppose a ~ b and b ~ c. Then a, b  A i for some i  I and b, c  A j for some j  I. That means that b  A i  A j. This is possible only if A i = A j. Therefore, a, c  A i. So, a ~ c.

28. Example Consider the set P of all computer programs. Partition P into subsets by putting in the same subset any two programs that always produce identical output for the same input.

29. Example This partition determines an equivalence relation  on P. Let p 1 and p 2 be two computer programs. Then p 1  p 2 if p 1 and p 2 always produce identical output for the same input.

30. Example Let A be the set of all people on Earth. Let R be the relation defined by x R y if x and y have ever shaken hands. Is R reflexive? Symmetric? Transitive? Let R * be the reflexive-transitive closure of R. Is R * an equivalence relation? What are the equivalence classes of R * ?