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 $R$.