Selected Problems on Equivalence Relations
11.3.3
Let \(A=\{a,b,c,d,e\}\). Suppose that \(R\) is an equivalence relation on \(A\) and \(R\) has three equivalence classes. Also \(aRd\) and \(bRc\). Write out \(R\) as a set.
11.3.7
Define a relation \(R\) on \(\mathbb{Z}\) as \(xRy\) if \(3x-5y\) is even. Prove that \(R\) is an equivalence relation and describe the equivalence classes.
11.3.13
Suppose that \(R\) is an equivalence relation on a finite set \(A\), and every equivalence class has the same cardinality \(m\). Express \(|R|\) in terms of \(m\) and \(|A|\).