Problem 1

Let \(F\) be a relation on the real numbers \(\mathbb{R}\), defined so that \((x,y)\in F\) if and only if \(x-y\in\mathbb{Z}\). Prove that \(F\) is an equivalence.

Problem 2

Consider the set \(A = \{1,2,3,4,5\}\) and let \(R\) be a relation on set \(A\), defined so that \((a,b)\in R\) if and only if \(|a-b|\) is even. Prove that \(R\) is an equivalence.

Problem 3

Let \(R\) be a relation on the natural numbers \(\mathbb{N}\), defined so that \((x,y)\in R\) if and only if \(y\) is divisible by \(x\). Determine if \(R\) is an equivalence. If it is, prove it. If it is not, give a set of counterexamples showing which properties of an equivalence do not hold for \(R\).