Are variables bounded to free variables still free variables?

Question

I am looking at some questions in my textbook about whether or not the variables are free or bound. I am not sure about these two in particular.

First off, I want to make sure I understand the concept of free vs. bound. I am fairly sure this x is a free variable in the following:

variable x is free in expression "x"

I believe this is true but I just want to make sure.

These two questions I am not really sure about, however.

(/ (+ 1 x) (let x 2 (+ x x))),
(let x y (/ (+ 1 x) (let x 2 (+ x x))))

For the top expression, the x in the first subexpression is unbound(right?) but x in the second subexpression is bound to 2, so would that mean the x in regards to the expression as a whole is unbound?

For the bottom expression, x is bound to y, but y is a free variable(?). So would x be free because y is free or is it bound because x is still bounded to y?

Answer 1

For (/ (+ 1 x) (let x 2 (+ x x))), the x in the first subexpression is unbound but x in the second subexpression is bound to 2, so would that mean the x in regards to the expression as a whole is unbound?

Yes. Although I would use the terminology "is bound" or "is free" only with respect to a concrete variable expression, not to a name. As you can see, it's ambiguous what "x" refers to.

I'd say "the whole expression has a free variable x", which is what you usually care about when trying to evaluate the expression.

For (let x y (/ (+ 1 x) (let x 2 (+ x x)))), x is bound to y, but y is a free variable. So would x be free because y is free or is it bound because x is still bounded to y?

The x is bound (and can be substituted). y is free.