8 Equivalence Relations 8.1 Relations 1. for sets A and B we define a relation from A to B to be a subset of A × B 2. for example the graph of a function is a relation from the domain (A) to its range (B). Note that every function a relation because it relates an element of A to an element of B. However not every relation a function because a relation may relate an element of A to more than just one element of B (or also because the empty relation is not a function). 3. a relation on a set A is a relation from A to A (i.e. subset of A × A, where A could be finite or infinite) 4. for a finite set A (with |A| = n), there are 2|A×A| = 2n2 possible relations on A, namely the elements of the powerset of A × A (note that the empty set is a relation on A as well) 8.2 Properties of Relations 1. a relation R defined on A is reflexive if ∀a ∈ A, then (a, a) ∈ R. 2. a relation R defined on A is symmetric if for elements a, b ∈ A we have that if (a, b) ∈ R then (b, a) ∈ R 3. a relation is transitive R defined on A if for elements a, b, c ∈ A we have that if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R 8.3 Equivalence Relations 1. a relation is an equivalence relation if it is reflexive, symmetric and transitive 2. if a relation R on a set A is an equivalence relation, then A can be partitioned into equivalence classes (note that the set A is partitioned, and the relation R helps find the partition, as the partition is unique for each relation) 3. if a relation R on a set A is an equivalence relation and a ∈ A, then the equivalence class [a] is given by [a] = {x ∈ A : xRa}, i.e. the equivalence class [a] contains a and all the elements related to a. 1 8.4 Properties of Equivalence Classes 1. if the class [a] has the same elements as the class [b], then the two classes are the same, i.e. [a] = [b]. It is always the case that two classes are either the same or they have no common elements (because the set of classes form a partition of A). This is Theorem 8.2. 2. since the equivalence classes produce a partition of A (see Theorem 8.3), we have the following: (1) no element of A belongs to two different equivalence classes, (2) every element of A belongs to an equivalence class, and (3) no equivalence class is empty. 3. also, the converse of the above item holds true: given a partition of a set A, there is always an equivalence relation R whose equivalence classes will be the elements of the partition. 8.5 Congruence Modulo n 1. recall divisibility definition: a|b if b = ak, ∃k ∈ Z 2. recall congruence modulo n definition: a ≡ b (mod n) if n|(a − b) 3. Theorem: Congruence modulo n is an equivalence relation on Z, and it has exactly n equivalence classes: [0], [1],..., [n − 1] (note that [0] = [n] = [2n] = ..., also [1] = [n + 1] = ...) 4. the classes above are given by the remainder of the given number when it’s divided by n 8.6 Integers Modulo n Skip this section. I will mention it briefly. 2.