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.

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.

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).

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

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}.

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.

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.

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.

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].

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

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.

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.

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).

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

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.

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)).

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

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)).

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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 * ?