10.3.1 Section 3: Equivalence Relations • Definition: Let R be a binary relation on A. R is an equivalence relation on A if R is reflexive, symmetric, and transitive. • From the last section, we demonstrated that Equality on the Real Numbers and Congruence Modulo p on the Integers were reflexive, symmetric, and transitive, so we can describe them as equivalence relations. 10.3.2 Examples • What is the “smallest” equivalence relation on a set A? R = {(a,a) | a Î A}, so that n(R) = n(A). • What is the “largest” equivalence relation on a set A? R = A ´ A, so that n(R) = [n(A)]2. Equivalence Classes 10.3.3 • Definition: If R is an equivalence relation on a set A, and a Î A, then the equivalence class of a is defined to be: [a] = {b Î A | (a,b) Î R}. • In other words, [a] is the set of all elements which relate to a by R. • For example: If R is congruence mod 5, then [3] = {..., -12, -7, -2, 3, 8, 13, 18, ...}. • Another example: If R is equality on Q, then [2/3] = {2/3, 4/6, 6/9, 8/12, 10/15, ...}. • Observation: If b Î [a], then [b] = [a]. 10.3.4 A String Example • Let S = {0,1} and denote L(s) = length of s, for any string s Î S*. Consider the relation: R = {(s,t) | s,t Î S* and L(s) = L(t)} • R is an equivalence relation. Why? • REF: For all s Î S*, L(s) = L(s); SYM: If L(s) = L(t), then L(t) = L(s); TRAN: If L(s) = L(t) and L(t) = L(u), L(s) = L(u). • What are the equivalence classes of R? • [q], [0], [00], [000], [0000], ... in other words, S0, S1, S2, S3, S4, ... 10.3.5 Relations and Partitions • Recall that a partition of a set is a collection of mutually disjoint subsets whose union is the original set. • Equivalence relations and partitions are tied together by the following: • Definition: Given a partition of a set A, the binary relation induced by the partition is R = {(x,y) | x and y are in the same partition set}. • Theorem: If A is a set with a partition and R is the relation induced by the partition, then R is an equivalence relation. 10.3.6 Making Equivalence Relations • This example shows how to apply this theorem to create the induced equivalence relation. • The collection {{1,2,3}, {4,5}, {6}} is a partition of {1,2,3,4,5,6}. To find the induced equivalence relation, observe: • {1,2,3} ´ {1,2,3} = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)} {4,5} ´ {4,5} = {(4,4), (4,5), (5,4), (5,5)} {6} ´ {6} = {(6,6)} • R = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (4,4), (4,5), (5,4), (5,5), (6,6)} 10.3.7 Formally Making Equivalence Relations • Theorem: Let A be a set partitioned by the collection {A1, A2, A3, ...}. Then the equivalence relation induced by the partition is given by: R = (A1 ´ A1) È (A2 ´ A2) È (A3 ´ A3) È ... • From the last example, the collection {{1,2,3}, {4,5}, {6}} partitioned {1,2,3,4,5,6}, so the induced relation is: R = {1,2,3}´{1,2,3} È {4,5}´{4,5} È {6}´{6} 10.3.8 Equivalence Classes and Partitions • Theorem: Let A be a set partitioned by the collection {A1, A2, A3, ...}. If a1ÎA1, a2ÎA2, a3ÎA3, etc., then the equivalence relation induced by the partition is given by: R = ([a1]´[a1]) È ([a2]´[a2]) È ([a3]´[a3]) È ... • From the last example, the collection {{1,2,3}, {4,5}, {6}} partitioned {1,2,3,4,5,6}, so the induced relation is: R = ([1] ´ [1]) È ([4] ´ [4]) È ([6] ´ [6]) 10.3.9 Going the Other Way • Theorem: Let A be a non-empty set and let R be an equivalence relation on A. Then the distinct equivalence classes of R partition A. • For example, given the relation of congruence mod 5 on the Integers, we obtain the partition: Z = [0] È [1] È [2] È [3] È [4]. • If S = {0,1}, what partition of S4 is induced by R = {(s,t) | s,t Î S4 and density(s) = density(t)}? S4 = [0000]È[0001]È[0011]È[0111]È[1111] 10.3.10 Functions, Relations & Partitions • Let f be a function defined on a set A, and consider the relation R={(a,b) | f(a) = f(b)}. Show R is an equivalence relation and describe the partition of A induced by R. • REF: f(a) = f(a) for all a Î A; SYM: If f(a) = f(b), then f(b) = f(a); TRAN: If f(a) = f(b) and f(b) = f(c), f(a) = f(c). • Each partition set contains those elements whose output from f is the same. 10.3.11 An Example • Let f be the function on Z, given by f(x) = x4 + 1. • x: 0 ±1 ±2 ±3 ±4 ... f(x): 1 2 9 82 257 ..., • This function induces the partition: Z = {0} È {-1, 1} È {-2, 2} È {-3, 3} È {-4,4}... 10.3.12 Graphs & Partitions • If we generate the directed graph of an equivalence relation just right, then the induced partition jumps out: 3 2 1 6 4 5.