Math 4310 Handout - Equivalence Relations

In class, we’ve been talking about the integers, which we’ve denoted Z. We started off talking about equality in Z, but then moved on to talking about congruences, which is a weaker notion. This handout explains how “congruence modulo n” is something called an equivalence relation, and we can use it to construct a set Z/nZ that’s a “quotient” of Z. The key point is that a congruence modulo n in Z becomes an equality in Z/nZ (and in abstract algebra, we really like to phrase things in terms of equalities). To start off, we need to say what we mean by a binary relation R on a set A. The idea is that any two elements a, b should be able to be compared, and either are “related” (which we might write aRb) or “not related” (which we can write aRb). To formalize this, we can define a binary relation to be any subset R ⊆ A × A, i.e. R can be any set of ordered pairs in A. If R is our relation, we say a and b are related if (a, b) ∈ R and they’re not related otherwise. To make it clearer what we mean, we usually use some symbol like ∼ to denote the relation, since a ∼ b looks much better than aRb to denote “a and b are related.” So, if ≡ (mod n) or ≡n denotes congruence modulo n, we can formalize this being a relation by saying it consists of pairs of integers (a, b) ∈ Z×Z such that n divides a− b. In fact, ≡n is a particularly nice type of relation called an equivalence relation: