# "such that" in logical statements

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?

• "For any natural number $x$ there is a number $n$ such that $n$ is the successor (i.e. $x+1$) of $x$" : $\forall x \exists n (n=x+1)$. Commented Sep 11, 2015 at 6:32

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.

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

$∀x \space ∃n, n=x^2$

- OR -

$∀x \space ∃n. n=x^2$

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