Question

Find the counter example, if possible to these universally quantified statements,where the domain for all variables consists of all integers.

Ax (x > 0 V x < 0)

Answer 1

Since for $x=0$ we get that $0<0$ is false and $0>0$ is false, we conclude that $(0>0)\lor (0<0)$ is false, and hence $\forall x(x>0\ \lor\ x<0)$ is also false. Therefore, the counter example is for $x=0.$