Originally Posted by

**SlipEternal** You change all "for all" signs to "there exists" signs, all "there exists" signs to "for all" signs, and negate any expressions based on those. So, let's negate emakarov's example: $\displaystyle (\forall x \in \mathbb{R})(\forall y \in \mathbb{R})(x < y \Rightarrow Q(x,y))$

It's negation would be: $\displaystyle (\exists x \in \mathbb{R})(\exists y \in \mathbb{R}) \neg (x<y \Rightarrow Q(x,y))$. So, how do you negate a conditional statement? The only time a conditional $\displaystyle A \Rightarrow B$ is false is when you have $\displaystyle A \wedge \neg B$. So, it would be $\displaystyle (\exists x \in \mathbb{R})(\exists y \in \mathbb{R})(x<y \wedge \neg Q(x,y))$