1 / 30

Equivalence Relations

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

sullivank
Download Presentation

Equivalence Relations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 M1, M2, and x0 such that M1g(x)  f(x)  M2g(x), for all x > x0.

  17. Example • It follows that (1/M2)f(x)  g(x)  (1/M1)f(x), for all x > x0. • 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 M1 and x1 and M2 and x2 such that f(x) M1g(x) for all x  x1 and

  19. Example g(x) M2h(x) for all x  x2. • Let x0 = max(x1, x2). • Then for all xx0, f(x) M1g(x)  M1  M2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 • [xa], aR, a > 0. • [bx], bR, b > 1. • [xa logbx], aR, a > 0, b > 1.

  22. Example • Furthermore, • [xa]  [xb] if a b. • [ax]  [bx] if a b. • However, • [logax] = [logbx] for all a, b > 1.

  23. The Equivalence Relation Induced by a Partition • Let A be a set and let {Ai}iI be a partition of A. • Define a relation ~ on A as x ~ yx, yAi 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 Ai 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Ai for some iI. • So b, aAi 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Ai for some iI and b, cAj for some jI. • That means that bAi Aj. • This is possible only if Ai = Aj. • Therefore, a, cAi. • 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 p1 and p2 be two computer programs. • Then p1  p2 if p1 and p2 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 xRy 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*?

More Related