Is $R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,(a - b)$ is an odd number $ \}$ an equivalence relation?

Question

$R \subset \Bbb N \times \Bbb N$

Is this an equivalence relation?

$R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,(a - b)$ is an odd number $ \}$

I say it is not because $(a, a)$ is always $0$ which is considered to be an even number.

Answer 1

It is not an equivalence relation because it fails to be reflexive as you noticed: $(a, a)$ is not in $R$ since $a-a = 0$ is even. Furthemore, it is not transitive since, for example $(2, 1) \in R$ and $(1,4) \in R$ but $(2,4) \not\in R$ since $2-4 = -2$.