$(10$ marks$)$ Consider the following proposition:

$p=$ "For all integers a and $b,$ if $a^2-b^2$ is even, then either $a-b$ is even or $a+b$ is even."

$($a$)$ Express the proposition $p$ as a predicate logic expression. In your answer you may use the predicate $Q(x)="x$ is even" or define your own primitive predicates.

$($b$)$ State the contrapositive of proposition $p.$ You may give this as a natural language statement or using predicate logic.

$($c$)$ Demonstrate that $p$ is true by proving the contrapositive. $,$You may use the fact that $(a+b+c{)}^2=a^2+b^2+c^2+2ab+2ac+2bc$