Am I understanding this first order formula correctly?

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The formula is:

$\forall x \; \exists y \; ( y < x \land \forall z (z < x \rightarrow \lnot y < z ))$

The way I understand it, it says for all $x$, there exists a $y$ such that $y < x$ and for all $z$, if $z < x$, then $y \le z$.

For the sets U = $\Bbb Z$ and U = $\Bbb R$, would this be true?

In the case y + 1 = x, that would be true, so there does exist a y. Or am I supposed to see if this is true for all y? (In which case it wouldn't be.)