I need some help with this question, what is the negation of

$\forall x >1$

would it just be,

$\exists x>1$ or $\exists x\le1$


  • 2
    $\begingroup$ The negation of "all $x$ are greater than $1$" must be: "there is an $x$ not greater than $1$". "Not greater" is "less or equal". $\endgroup$ Oct 13, 2017 at 12:32
  • $\begingroup$ But see also Restricted quantifiers. $\endgroup$ Oct 13, 2017 at 12:59

1 Answer 1


"For every $x$ greater than $1$, blah" is falsified by an example of an $x$ greater than $1$ such that blah is false.

So $(\forall x > 1)(P(x))$ is negated to $(\exists x > 1)(\neg P(x))$.

  • 1
    $\begingroup$ Another useful intuition is that a statement of the form $\forall x > 1,\, P(x)$ tells you nothing about the truth or falsity of $P(x)$ when $x \le 1$, so neither can its negation. $\endgroup$ Oct 13, 2017 at 14:59

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