Take \(A = \{0, 1, 2, 3, 4, 5\}\). Define a relation \(R\) on \(A\) by \(x\, R\, y\) if and only if \(x^2 - 4x = y^2 - 4y\).
Prove that \(R\) is an equivalence relation.
Exhibit the partition of \(A\) whose members are the equivalence classes of \(R\).
Solution
We need to show that \(R\) is reflexive, transitive, and symmetric. 1. Reflexive: this would say that \(x\sim x \iff x^2-4x = x^2-4x\), which is true. 2. Transitive: suppose \(x\sim y\) and \(y\sim z\), we want to show \(x\sim z\). But we have \[ x^2 - 4x = y^2-4y ~\&~ y^2-4y = z^2-4z \implies x^2-4x = y^2-4y = z^2-4z \] 1. We want to show \(x\sim y \implies y \sim x\), which follows because \(x^2-4x = y^2-4y \iff y^2-4y = x^2-4x\). 2. The equivalence classes: \[\begin{align*} x^2-4x &= 0: &\theset{0, 4}\\ x^2-4x &= -3: &\theset{1,3} \\ x^2-4x &= -4: &\theset{2} \\ x^2-4x &= 5: &\theset{5} \end{align*}\]