0
$\begingroup$

I have a simple question about the usage of "such that" in logical statements.

Let the statement S be: "For every $x \in A, f (x) > 5$."

In the negation of S:

There is an $x \in A$ such that $f (x) ≤ 5$.

The comma in S is converted to "such that" in the negation of S.

Can somebody clarify on the "such that" came in the negation of S.

$\endgroup$
  • 1
    $\begingroup$ Usually you use "such that" in combination with $\exists$ (exists). Since to negate $\forall$ you use $\exists$, it's natural to use "such that" as well. $\endgroup$ – rubik Feb 20 '15 at 15:10
2
$\begingroup$

Rubik is right: There is an implied "such that" in the first statement as well.

For every $x\in A$, $f(x)>5$ means $\forall\,x$ such that $x\in A$, $f(x)>5$, which in turn can be translated to $\forall\,x\quad x\in A\implies f(x)>5$.

$\endgroup$
  • $\begingroup$ And likewise, the negation: There is an x such that ($x\in A$ AND $f(x) \leq 5)$. (This isn't directed at you, @Tim, nor am I insinuating and criticism. I'm just adding an observation (like you suggest) that such that appears often in the parsing of quantified statements.) $\endgroup$ – Namaste Feb 20 '15 at 15:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.