Give a counterexample, if possible, to this universally quantified statements, where the domain for all variables consists of all integers. $∀x(|x|>0)$.
I think the counterexample is as simple as $∀x(-|x|>0)$ but i am having doubts. What am i missing?
Best Answer
To give a counter example for a statement that starts with "For all," you only need one object for which the statement does not hold. In this case, that object is 0. Since $|0|=0$ and $0$ is an integer, the statement "$\forall \text{ integer } x: |x|>0$" is wrong.