## Bound and Free Variables [closed]

Does a variable in an expression need to be explicitly bound by a quantifier (or by some other operator), to be bound? Or can it be implicitly bound? For example:

If $x = y^2$ and $y > 0$, then, for any number $z$, $x > -z^2$.

This is sentence is universally satisfied for any x and y adhering to the given restraints, and therefore doesn't depend on x or y. Yet they are not specifically bound by a quantifier. Are x and y free or bound?

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 Well, this sentence is equivalent to $\forall z \in \mathbb{R}, -z^2 < y^2$ (where $y\in \mathbb{R}$). – David Roberts Jan 18 2011 at 4:30 I'm not sure the quantization tag is appropriate... – Yemon Choi Jan 18 2011 at 4:55

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