How to express that a pair in a set satisfies a condition

Question

Simple question, but how would I express that for any ordered pair $n$ in set $M$, $n$ is "special" if its pair adds up to $4$? E.g. $(0,4), (1, 3), (2, 2), (3, 1)$. Would this example be appropriate?

$M = \text{a set of ordered pairs}$

$f(n) : n\text{ is special}$

$$\exists n \in M, n_{x} + n_{y} = 4 \Rightarrow f(n)$$

Answer 1

Almost: Let $M$ be a set of ordered pairs of natural numbers i.e. $M \subseteq \mathbb{N} \times \mathbb{N}$. Then define for any $n = (n_x, n_y) \in M:$

$$f(n) :\Leftrightarrow n ~ \text{is special} ~.$$

Then we have

$$ \forall n \in M: f(n) \Leftrightarrow n_x + n_y = 4$$

Note the logical equivalence $\Leftrightarrow$ in the last line.