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.
The question is asking you to use the rules of deduction you have been given to show that the two sides of the equivalence are exactly the same as each other - so in this case, show that "((for all x: P(x) is true) or (A is true))" is logically equivalent to "for all x: (P(x) is true or A is true)".
"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.)