Writing a simple expression in set notation I am trying to state that for every pair of even integers, the sum of the two integers is even.
$\forall m, n \in 2 \mathbb Z, \exists a = m + n \ni a \in 2 \mathbb Z$ 
Does this make sense?
 A: What you wrote is comprehensible and has the correct meaning, but it is expressed kind of strangely and there is a much simpler way to write it.  In English, what you wrote is

For all even integers $m$ and $n$, there exists $a$ which is equal to $m+n$ such that $a$ is even.

The second half of this sentence is a pretty convoluted way to say "$m+n$ is even"!  There is no need to make a variable $a$ and say "there exists $a$": $m+n$ is a specific number, not an unspecified number whose existence you are asserting.  So instead you can just say 

$$\forall m,n\in 2\mathbb{Z}, m+n\in 2\mathbb{Z}.$$

Or in English:

For all even integers $m$ and $n$, $m+n$ is even.

A: While your expression is logically true, according to your statement, it's better to say:
$$\forall m, n \in 2 \mathbb Z, m + n \in 2 \mathbb Z$$
A: The symbol $\exists$ is redundant. You could also state: 
$$\forall m,n\in 2\mathbb{Z} \ \ and \ \ k=m+n, \ \ k\in2\mathbb{Z}$$
A: 
Another variant:
\begin{align*}
2\mathbb{Z}+2\mathbb{Z}\subseteq2\mathbb{Z}
\end{align*}

Note: The following is valid
\begin{align*}
2\mathbb{Z}&=\{2a: a\in \mathbb{Z}\}\\
2\mathbb{Z}+2\mathbb{Z}&=\{2a+2b: a,b\in \mathbb{Z}\}\\
\end{align*}
