Question

To prove there is a function \(f\) with A as its domain such that \((x,y)\) ? \(R\) if and only if \(f(x) = f(y)\).

Short answer

\(R\) is an equivalence relation.

Step-by-step solution

Given data

A is a nonempty set and \(R\) is an equivalence relation on A.

Concept used of equivalence relation

An equivalence relation is a binary relation that is reflexive, symmetric and transitive.

Prove equivalence relation

First fix an arbitrary representative for each of the equivalence classes of \(A\) under \(R\).

Let us define a function f as \(f:A \to A \ni f(x) = {x_0}\), where \({x_0}\) represents \({(x)_R}\forall x \in A\).

\(\begin{array}{l}(x,y) \in R\\{\rm{ If }}x,y \in {(x)_R}\quad \\f(x) = {x_0} = f(y)\\f(x) = f(y)\\{x_0} = {y_0}\\{\rm{If }}{(x)_R} = {(y)_R}\quad \\y \in {(x)_R}\\(x,y) \in R\end{array}\)

The last step follows as \(R\) is an equivalence relation.