<s>[INST] Prove or disprove that where $a, b \in Z$, the number a+b is even if and only if a-b is even. 
Prove or disprove that where $a, b \in Z$, the number a+b is even if and only if a-b is even.

Can some help me improve on this or check it please?
If $a + b$ is even, then there is some number $k \in Z$ such that $a + b = 2k$.
If $a - b$ is odd, then there is some number $m \in Z$ such that $a - b = 2m +1$.
$a + b = a - b + 2b = 2m + 1 + 2b = 2(m + b) +1$. This is an odd number.
Now try for $a - b$ is even.
If $a - b$ is even, then there is a number $m \in Z$ such that $a - b = 2m$.
$a + b = a - b + 2b = 2b + 2m = 2(b + m)$. This is an even number.
Therefore, $a + b$ is even if and only if $a - b$ is even.
 [/INST] We have $a+b$ even iff $a+b=2k$ for some $k$ iff $a+b-2b = 2k-2b=2(k-b)$ even iff $a-b=2(k-b)$ even.
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