Question

To determine \(\left( {\frac{1}{2}} \right)\), the equivalence class of \(\frac{1}{2}\), for the given relation \(R\).

Short answer

Equivalence class ,.

Step-by-step solution

Given data

The relation \(R\) defined on the set of real numbers by

\(R = \{ (x,y):x - y{\rm{ is aninteger }}\} \).

Formula used of Equivalence class

An equivalence class is defined as a subset of the form\(\{ x \in X:xRa\} \), where\(a\)is an element of\(X\)and the notation "\(xR{y^{\prime \prime }}\)is used to mean that there is an equivalence relation between\(x\)and\(y\).

Find the equivalence class

By definition of an equivalence class,

This gives, .