How to prove this equivalence relation?

Question

How would one go about proving this is an equivalence relation? I have no idea where to start.

$\cal R$ is the relation on $\Bbb Z \times \Bbb Z$, such that $((a, b),(c, d)) \in \cal R$ if and only if $a−d = c−b$.

Answer 1

If function $f:\mathbb Z\times\mathbb Z\rightarrow\mathbb Z$ is prescribed by $(a,b)\mapsto a+b$ then $$((a,b),(c,d))\in\mathcal R\iff f(a,b)=f(c,d)$$ This makes it easy to prove that the relation is reflexive, symmetric and transitive:

• $f(a,b)=f(a,b)$
• $f(a,b)=f(a',b')\Rightarrow f(a',b')=f(a,b)$
• $f(a,b)=f(a',b')\wedge f(a',b')=f(a'',b'')\Rightarrow f(a,b)=f(a'',b'')$