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.

  • 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

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 $\forall\,x\quad x\in A\implies f(x)>5$.

  • $\begingroup$ 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

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