Question

To determine the equivalence classes under the given relation\(R\).

Short answer

There is only one equivalence class, as all vertices are related to each other.

Step-by-step solution

Given data

The relation \(R\) on the three vertices of an equilateral triangle \(A,B\) and \(C\). A vertex \(X\) and \(Y\) are related if \(Y\) can be obtained from \(X\) either by a rotation and a rotation followed by a reflection.

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, the equivalence class of any vertex \(X\) is the set of all vertices which are related to \(X\) under \(R\).

Under the given relation any vertex can be obtained from the other by means of a rotation. Thus \((A) = (B) = (C)\).

So, there is only one equivalence class.