$($a$)($Graded for correctness ${?}^1)$ Express the statements $($i$)-($v$)$ as quantified statements. Define any predicates as needed.

$($b$)($Graded for correctness of choice and fair effort completeness in justification ${?}^2)$ What are the error$($s$)$ in the following attempted proof for statement $($ii$)?$

We prove this by contrapositive. So$,$ by universal generalization let $x$ and $y$ be arbitrary integers, such that $x$ and $y$ are not both even. Without loss of generality, suppose that $x$ is even and $y$ is odd. Then $x=2k$ for some integer $k$ and $y=2j+1$ for some integer $j.x+y=2k+2j+1=2(k+j)+1.$ Since $j$ and $k$ are both integers, $j+k$ is too. Therefore, $x+y$ is odd. This proves the contrapositive, thus proving the claim.

$($c$)$ Write the correct version of the proof for statement $($ii$)$ if it is true or provide a counterexample to disprove it if it is not.

$($d$)($Graded for correctness of choice and fair effort completeness in justification$)$ Which of the statements $($i$)-($v$)$ is being disproved by the following proof?

To disprove the statement, we need to find a counterexample. We need to show that there exist some $x$ and $y$ in the domain such that