# Discrete Mathematics - Quantifiers problem

This is a question from the Discrete Mathematics question from Kenneth Rosen book.

I didn't understand the question and thus I am confused how to begin with question. Below is the question from the book.

Establish these logical equivalences, where x does not occur as free variable in A. Assume that the domain is nonempty.

a) ( ∀x P(x)) ∨ A) ≡ ∀x (P(x) ∨ A)

Also what does "x does not occur as free variable in A" mean. Thank You.

• Hint: To get a better understanding of what is going on try to to show this equivalence for the following 2 particular cases: the case where $A$ is $\forall x R(x,y)$ and the case $\exists x R(x,y)$. In other words, start checking that $(\forall x P(x)) \lor ( \forall x R(x,y) ) \equiv \forall x (P(x) \lor \forall x R(x,y) )$ and that $(\forall x P(x)) \lor ( \exists x R(x,y) ) \equiv \forall x (P(x) \lor \exists x R(x,y) )$ – boumol Jul 13 '16 at 7:26

"x does not occur as a free variable in A" means that you can assume that A is not affected by the value of x, because if you were to write A out as a statement you wouldn't find x appearing as something whose value can be varied. (In P(x), x is a free variable, but in $\forall x P(x)$, x is "bound" by the $\forall$ qualifier.)