Question

Let R be an equivalence relation on Z, which has P={{±i}|i ∈ N} as its collection of equivalence classes. Describe the equivalence relation R.

Answer 1

By definition, $(a,b)\in R$ if and only if $a,b\in A$ for some $A\in P.$

Let us describe the equivalence relation $R$. It follows that for $i\in\mathbb N$ we have that $a,b\in\{\pm i\}$ if and only if $|a|=|b|=i$ . Therefore, $a,b\in\{\pm i\}$ for some $i\in\mathbb N$ if and only if $|a|=|b|$ .

               We conclude that $(a,b)\in R$ if and only if $|a|=|b|$.