Question about free and bound variables Which of the variables $x$ and $y$ are free and which are bound?
$\forall x : P (x, y) \to (\exists y : Q(x, y) \lor \exists x : R(x))$
The solution says that $y$ is free and both $x$ and $y$ are bound. But why isn't $x$ a free variable?
 A: 
$1.$ In logic a varible is free, if and only if it does not occur in the scope of a quantifier.
$2.$ A varible is bounded, if and only if it occurs in the scope of a quantifier.

That we have:
$$\color{orange}{∀x} \underset{\color{orange}{\text{bounded by x}}}{\underline{( P (\color{orange}x, \underset{\color{grey}{\text{free}}}{\underline{\color{grey}y}}) → (\underset{\color{red}{\text{bounded by y}}}{\underline{\color{red}{\color{red}{∃y}}~Q(\color{orange}x, \color{red}y)}} ∨ \underset{\color{blue}{\text{bounded by x}}}{\underline{\color{blue}{∃x}~(R\color{blue}x)}})}})$$
And I believe this is what you thought: $$\color{orange}{∀x} \underset{\color{orange}{\text{bounded by x}}}{\underline{( P (\color{orange}x, \underset{\color{grey}{\text{free}}}{\underline{\color{grey}y}})) }}→ (\underset{\color{red}{\text{bounded by y}}}{\underline{\color{red}{\color{red}{∃y}}~Q(\color{grey}x, \color{red}y)}} ∨ \underset{\color{blue}{\text{bounded by x}}}{\underline{\color{blue}{∃x}~(R\color{blue}x)}})$$
So we also have $x$ free in $Q(x,y),$ which is not what the question talking about $\dots$
A: A very confusing example. 
HINT: If the author's intent was to MINIMIZE confusion, they should have written something like: 
$\forall a: P(a,y) \implies (\exists b: Q(x,b) \lor \exists c: R(c))$ 
Here, $a$, $b$ and $c$ are bound variables; $x$ and $y$ are free variables.
Tip: I like to use the letters a-k for bound variables, the rest for free variables. If possible, you should  avoid using the same variable as both free and bound in the same statement.
