Prof of Reflexive, symmetric, or transitive relations

Question

Consider the relation R on Z as: ∀m,n ∈Z, mRn ⇔ m − n is odd . Is R reflexive, symmetric,
or transitive? What would the proof or counter proof be?

Since R is a reflexive since m-n is linear, but I'm not sure how that would work with the proofs.

Answer 1

Update: it is not true, but it is true for if the relation is $mRn \Leftrightarrow m-n$ is even

Let's prove the three properties:

Reflexive: is it true that $mRm$ for all $m$?

Yes, $m-m=0 \Rightarrow mRm$, and 0 is even.

Symmetric: if $mRn$, then $nRm$?

$mRn \Rightarrow m-n$ is even $\Rightarrow n-m$ is even (the opposite of an even is another even) $\Rightarrow nRm$

Transitive: if $mRn$ and $nRp$, is it true that $mRp$?

We have $m-n$ and $n-o$ are both even numbers. Now $m-p = (m-n)+(n-o)$. Just realise that even plus even is even, so $m-p$ is even and therefore $mRp$

I think with this you can see why it is false for odd numbers.