# for all x less than or greater than a number

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

$\forall x >1$

would it just be,

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

Thanks

• 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". Oct 13, 2017 at 12:32
"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 $(\exists x > 1)(\neg P(x))$.
• 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. Oct 13, 2017 at 14:59