Give a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. That is, show a reason why the statement is NOT universally true when applied to the domain of integers.
a. $\forall x (|x| > 0)$
b. $\forall x \exists y (x = 1/y) $
c. For each of the quantified statements in a-b above, give a domain for the variables for which each universally quantified statement a-b is true.
For part a I put $x=0$. For b. I'm not sure what it is asking. I put $y=0$ as a shot in the dark but I've no clue.
Best Answer
a. Let $x=0$. Then $|x|=0\not> 0$.
b. Let $x=2$. Then there does not exist a $y\in\mathbb Z$ such that $x=\frac{1}{y}$.
c. The first statement is satisfied $\forall x\in\mathbb Z\setminus\{0\}$ since $\forall x\in\mathbb Z\setminus\{0\}, |x| > 0.$ The second statement is only satisfied $\forall x\in\{-1,1\}$ because $1=\frac{1}{1}$ and $-1=\frac{1}{-1}$ (observe that the domain of $x$ would be $x\in\mathbb Z\setminus\{0\}$ if we allowed $y\in\mathbb R$).
Therefore, the domain for the first statement is $x\in\mathbb Z\setminus\{0\}$ and the domain for the second statement is $x\in\{-1,1\}$