Section 1.4: Composition of Functions

Solution
Clearly, \((x, y) R (x, y)\), \(\forall (x, y) \in A\), since \(xy = yx\).
This shows that \(R\) is reflexive.
Further, \((x, y) R (u, v) \Rightarrow y u = v x\) and hence \((u, v) R (x, y)\).
This shows that \(R\) is symmetric.
Similarly, \((x, y) R (u, v)\) and \((u, v) R (a, b) \Rightarrow y u = v x\) and \(v a = u b \Rightarrow a u = y a\) and hence \((x, y) R (a, b)\).
Thus, \(R\) is transitive.
Thus, \(R\) is an equivalence relation.