Answer to Question #76317 in Discrete Mathematics for Chad

Question

Let X be a non-empty set, and let R be an equivalence relation on X. Let C be the set of all equivalence classes of R. So C={A⊆X such that A=[x] for some x ∈ X}.

Now, define f : X → C by the rule f(x) = [x] for all x ∈ X.

Prove that if x ∈ X, then there is one and only one equivalence class which contains x.

Suppose X = {1, 2, 3, 4, 5} and that R is an equivalence relation for which 1 R 3, 2 R 4 but 1 R̸ 2,1 R̸ 5,and 2 R̸ 5.

Write down the equivalence classes of R and draw a diagram to represent the function f.