Prove that R is an equivalence relation.

Question

Let $A$ be the set of integer ordered pairs.

Define relation $R$ on $A$ by $(a,b)R(c,d)$ $\iff$ $a+d = b+c$.

Prove that $R$ is an equivalence relation.

Answer 1

• Reflexive

$a+b=b+a\implies (a,b)$R$(a,b)$

• Symetric

$a+d=b+c\implies c+b=d+a$ $\implies (c,d)$R$(a,b)$

• Transitive

$a+d=b+c$ and $c+f=d+e$ $ \implies a+d+c+f=b+c+d+e$

$\implies a+f=b+e\implies$ $(a,b)$R$(e,f)$.

Answer 2

If you can find a function $f$ such that $f(a,b) = f(c,d)$ iff $(a,b)R(c,d)$ then you can show that $R$ is an equivalence relation (because, in some sense, you have reduced to the problem to a known equivalence relation $=$).

The equivalence relation in the question suggests a fairly straightforward $f$.