free variable predicate logic

Question

I'm trying to understand the concept of free variables in predicate logic.

To my understanding, in the equation:

∃x.∀y.x ≥ y

x is not a free variable (it is bound) since it is within the scope of ∃x

Answer 1

Your understanding seems to be correct (though I wouldn't call the line you displayed an "equation", rather a "formula" or, in this case, "sentence").

Strictly speaking, one ought to describe an instance of a variable as being free or bound, because the same variable could appear both ways in a formula. For example, the variable $x$ in $(x > 0) \wedge (\exists x, x < 5)$ appears free (the first time it appears) and bound (the second time it appears). Unless you're working in a very specific context where your supply of variables is limited, it's more humane to re-write the above expression as $(x > 0) \wedge (\exists y, y < 5)$, where now $x$ appears free and $y$ appears bound. Some authors expressly prohibit having the same variable appear both free and bound in the same formula.