Let $R$ be the relation on $\mathbb Z^+ \times \mathbb Z^+$ such that $(a, b)R(c, d)$ if $gcd(a, b) = gcd(c, d)$?

Question

I need to find out:

1. Prove that $R$ is an equivalence relation. (I am not clear on definition of an equivalence relation)

2. What is the equivalence class of $(1,2)$?

3. Give an interpretation of the equivalence classes of $R$.

This as far as I could go, see below.

I am wondering if I got it right this time?

Answer 1

A relation $\mathcal R$ is an equivalence relation if it's reflexive, symmetic and transitive.

The equivalence class $[(1,2)]=\{(a,b)\in\mathbb{Z^+}\times \mathbb{Z^+}\ | \gcd(a,b)=1$} hence $a$ and $b$ are coprime.

Notice that any relation $\mathcal R$ defined by equality: $$x\mathcal R y\iff f(x)=f(y)$$ is arguably an equivalence relation.