Define a relation R on Z × N by (a, α)R(b, β) if and only if aβ = bα. Prove that R is a reflexive relation.

Question

I'm a bit confused about how to prove that R is reflexive.

By definition, R, a relation in a set S, is reflexive if and only if ∀x∈S, xRx.

Since (a, α)R(b, β), we know that aβ = bα.

Then to prove that this is reflexive, based on the definition, we would have to show that ((a, α)R(b, β)) R ((a, α)R(b, β). After this, I'm not sure as to how to prove why this is reflexive.

Could we possibly do something like (aβ = bα) R (aβ = bα) is reflexive? Or does ((a, α)R(b, β)) R ((a, α)R(b, β)) already show that it is reflexive itself?

Answer 1

To prove that $R$ is reflexive is to prove that, for each $(a,\alpha)\in\Bbb Z\times\Bbb N$, $(a,\alpha)\mathrel R(a,\alpha)$. But $(a,\alpha)\mathrel R(a,\alpha)$ means that $a\alpha=a\alpha$, and it is clear that this holds.