"such that" in logical statements

Question

How exactly do I put "such that" into logical statements?

For any $x,$ there exists an $n$ such that $P(x).$

I know to start with $∀x ∃n,$ but where do I go from here?

Answer 1

How exactly do I put this into logical statements?

For any x, there exist an n such that P(x).

I know to start with ∀x ∃n, but where do I go from here?

You add $P(x)$

$$\forall x\exists n\;P(x)$$

A space typeset between the n and P improves legibility.   Sometimes this may not seem enough, so you might write $\forall x\exists n{:}P(x)$ or $\forall x\exists n{.}P(x)$ or use some other punctuation mark to clearly distinguish between the quantified variables and the predicate bound by them (I prefer the colon).   These symbols are optional; they're just to add clarrity.   Parenthesis do the same work.

$$\forall x\exists n\,[P(x)]$$

Answer 2

$∀x \space ∃n, n=x^2$

- OR -

$∀x \space ∃n. n=x^2$

"such that" is commonly represented simply by a comma or a dot.