“such that” in logical statements

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?

• "For any natural number $x$ there is a number $n$ such that $n$ is the successor (i.e. $x+1$) of $x$" : $\forall x \exists n (n=x+1)$. – Mauro ALLEGRANZA Sep 11 '15 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.

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

$|$ symbol is used to denote "such that." Ex:

$\{x \ | \ x \ mod \ 2 \ \text{is } \ 0\}$

The above line can be interpreted as "The set of all values of $x$ such that $x$ when divided by $2$ yields $0$ as the remainder."

• I’ve never seen that used with existential quantifiers. – amd Sep 11 '15 at 6:41
• This is usable only for statements that involve "For all" rather than "there exists" – Deepak Gupta Sep 11 '15 at 6:47
• Oops, this is used in set builder form. – Shubham Sep 11 '15 at 6:47

$∀x \space ∃n, n=x^2$

- OR -

$∀x \space ∃n. n=x^2$

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