Discrete Math quantifiers,proof and demonstrating with symbols

Question

In my Discrete Math course, I encounter a question such as $$\exists x\forall y\,x<y \text{ where }x,y\in\mathbb{Z}\,,$$ and I explain it like this: "Let $x=1$ and $y=-2$ For an $x$ in the universe not all $y$ values make the statement $x < y$ true so the case is $1$ is not equal $-2$" and I think this is sufficient for explaining but I wonder whether there is a way to indicate proof with symbols or mathematically. Thanks in advance.

Answer 1

With quantifiers, you can't choose the specific values in all cases. In particular:

• You can choose a specific value for an $\exists$-quantified variable to prove a statement true, but not to prove it false.
• You can choose a specific value for a $\forall$-quantified variable to prove a statement false, but not to prove it true.

What you've done is proved that there is a particular $x$ that the statement doesn't hold for. But that doesn't mean that there isn't another value that it does hold for! (Or at least, it doesn't inherently mean that). If you're showing there doesn't exist an $x$, you need to show that the statement $\forall y \ldotp x < y$ is false for every single $x$, not just a specific $x$.