A correct logical representation for an iff expression For every integer n, $n^{3}$ is even if and only if n is even
This is clearly an implication, the problem is the order of the statement confuses me.
$n^{3}$ is even $\Longrightarrow$ n is even
vs
n is even $\Longrightarrow$ $n^{3}$ is even
 A: If and only if means they both imply each other, sometimes denoted $P \iff Q$, $P$ if and only if $Q$.
That is, both:
$n^{3}$ is even $\implies$ $n$ is even
and
$n$ is even $\implies$ $n^{3}$ is even
When an if and only if statement is true, the statements are said to be equivalent. In your case, the condition that for a natural number $n$, $n^3$ is even, is equivalent to $n$ being even. 
A: This is more than an implication. It is a bidirectional implication. If you want purely symbolic notation, then you might write this as
$$\forall \, n \in \mathbb{Z}, (\exists j\in \mathbb{Z},\,n^3 = 2j) \iff (\exists k\in \mathbb{Z},\,n = 2k),$$
and that could probably be refined. I expect purely symbolic notation is not critical here. In essence you need the $\iff$ in your statement. 
If you want to prove this, then the burden is to assume only the left side and then prove the right, then assume only the right side and then prove the left.
As a logical expression that most closely demonstrates the answer to your question,
$$n^3 \, \text{is even} \iff n \, \text{is even}$$
should suffice.
While proving that $n \, \text{is even} \Rightarrow n^3 \, \text{is even}$ should seem trivial, proving that $n^3 \, \text{is even} \Rightarrow n \, \text{is even}$ is usually done by proving the contrapositive. That will take bit more thought.
