3. R = = on Z. R, S, A, T.

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1 6 Relations Let R be a relation on a set A, i.e., a subset of AxA. Notation: xry iff (x, y) R AxA. Recall: A relation need not be a function. Example: The relation R 1 = {(x, y) RxR x 2 + y 2 = 1} is not a function. Some definitions 1. R is reflexive iff xrx x A. 2. R is symmetric iff xry yrx x, y A. 3. R is antisymmetric iff xry yrx x = y x, y A. 4. R is transitive iff xry yrz xrz x, y, z A. We now give a number of examples. 1. R = < on Z. R, S, A, T. 2. R = on Z. R, S, A, T. 3. R = = on Z. R, S, A, T. 76

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4 Equivalence Relations Definition: A relation R AxA is an equivalence relation if it is reflexive, symmetric and transitive. Examples: 1. = on Z. 2. The universal relation U A = AxA, i.e., the relation consisting of all elements of AxA. 3. Let A be the set of all triangles in the plane. Then T 1 RT 2 iff T 1 and T 2 are similar triangles. 4. Let A be the set of all points in the plane. Then p 1 Rp 2 iff the distance from p 1 to the origin equals the distance from p 2 to the origin. 5. A = Z, m Z, m > 0. ar m b iff m a b, i.e., c Z such that m c = a b. (a) R m is reflexive since m a a. (b) R m is symmetric since m a b m b a. (c) R m is transitive since if m a b and m b c, then m (a b) + (b c) or m a c. Notation: If m a b we say a is congruent to b mod m, or a b (mod m). 79

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6 Note: A partition of a set defines an equivalence relation in a very natural way. Definition: Let P be a partition of a set A. Then the equivalence relation R(P) associated with P is given by: ar(p )b iff a and b are in the same set in P. Note: R(P ) is clearly an equivalence relation. Example: The sets A 1, A 2, A 3 partition Z. A 1 = {... 6, 3, 0, 3, 6,...} A 2 = {... 5, 2, 1, 4, 7,...} A 3 = {... 4, 1, 2, 5, 8,...} Thus 1R(P)7 and -4R(P)8. We now wish to show that each equivalence relation on a set A defines a partition in a natural way. 81

7 Recall [b] = {x A xrb}. Theorem 6.3 Let R be an equivalence relation on a set A. Then 1. b [b] b A. 2. a, b A, [a] = [b] arb. 3. a, b A, either [a] = [b] or [a] [b] = φ. Note: Let P be the set of all partitions on a set A and E be the set of all equivalence relations on A. There is a 1 1 onto function f : P E given by f(p ) = R(P ). The function is clearly 1 1. To see that it is onto, start with any equivalence relation E on A. The partition that maps to it is the one obtained from the above Theorem, i.e., the partition obtained from the distinct equivalence classes of the elements of A. Example: Let s define Q. First, let F = {a/b a, b Z, b 0}. Define R F xf by a/b R c/d iff ad = bc. Hence 2/3 R 6/9. Also, [1/2] = {1/2, 2/4, 3/6, 4/8,...}. Thus each rational is an equivalence class in F under R. 82

8 Posets Let R be a relation on a set A. Then R is a partial ordering on A if R is 1. reflexive 2. antisymmetric 3. transitive Examples: 1. R : on Z. 2. R : on P(A), the power set of A. 3. R : divides on Z +. (a) a a. (b) a b and b a a = b. (c) a b and b c a c. 4. Let Σ be an alphabet with a partial ordering. Then the lexicographic (alphabetcal) ordering R on Σ is a partial ordering (see definition on p.636). Example: Σ = {a, b}. aab R aabab baa R bab e R baab aabb R b 83

9 Definition: If R is a partial ordering on A we call (A, R) a partially ordered set or poset. Notation: When the relation is a partial ordering, we often use a b instead of arb. Definition: Suppose (A, R) is a poset. Elements a and b of A are said to be comparable if, and only if, either arb or bra. Otherwise they are noncomparable. Definition: Let R be a partial order relation on a set A. If any two elements a and b in A are comparable, then R is a total order relation on A. 84

10 Examples: 1. R : on Z is a total order. 2. R : on P (A) is not a total order if A has more than 1 element. 3. R : divides is not a total order on Z +, e.g., 3 does not divide 5 and 5 does not divide R : lexicographic (alphabetic) ordering on Σ, where Σ is an alphabet with a partial ordering, is a total order. (See definition on p. 636). Definition: Let (A, R) be a poset. A subset B of A is called a chain if, and only if, each pair of elements in B is comparable. The length of a chain is the number of elements in the chain. Note: The book has a different definition of length. Example: The set P ({a, b, c}) is partially ordered with respect to subset inclusion. The set S = {φ, {a}, {a, b}, {a, b, c}} is a chain of length 4 in P ({a, b, c}). 85

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12 Definition: Let (A, ) be a poset. An element a A is a maximal element if there does not exist b A such that b a and a b. Note: minimal element is defined similarly. In the examples above a is a maximal element g is a minimal element 1, 2, 7 are maximal elements 8, 9, 10 are minimal elements Definition: A subset of a poset (A, R) is an antichain if no two distinct elements of the subset are related. Example: {c, f, e} Theorem 6.4 Let (A, ) be a poset. If n is the length of a longest chain in (A, ), then A can be partitioned into n disjoint antichains. Proof: Later, by induction. 87

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17 Theorem 6.6 Let R 1 AxB, R 2 BxC and R 3 CxD. Then (R 1 R 2 )R 3 = R 1 (R 2 R 3 ), i.e., the composition of relations is associative. Proof: ( ) Let (a, d) (R 1 R 2 )R 3. Then c C such that (a, c) R 1 R 2 and (c, d) R 3. Since (a, c) R 1 R 2 b B such that (a, b) R 1 and (b, c) R 2. Now (b, c) R 2 and (c, d) R 3 (b, d) R 2 R 3. But now (a, b) R 1 (a, d) R 1 (R 2 R 3 ). ( ) Similar. Definition: Let R be a binary relation on a set A. Then for all integers n 0, R n is defined as follows: 1. R 0 = {(x, x) x A}. 2. R n+1 = R n R. 92

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19 Theorem 6.7 Let A = n and R AxA. Then s, t, 0 s < t 2 n2, such that R s = R t. Proof: First note that AxA has n 2 elements. Hence there are 2 n2 distinct relations on A. By the pigeonhole principle, at least two of them are equal. 94