For each of the following rename the quantifiers to enforce

For each of the following, rename the quantifiers to enforce uniqueness of variables, migrate quantifiers to the front, remove the quantifiers, and finish the proof using equational reasoning (or explain why no such proof can be found). ((Forall x) P(x)) rightarrow ((x) P(x)) ((y) (Forall x) P (x, y)) rightarrow ((Forall x) (y) P(x, y) ([(Forall x) (P(x) rightarrow Q(x))] [(Forall x) (Q(x) rightarrow R (x)))]) rightarrow [(Forall x) (P x) rightarrow R(x)] [(x) (P(x) Q(x))] rightarrow [((x) P(x))) ((x) Q (x))] [((x) P (x)) ((x) Q(x))] rightarrow [(x) (P(x) Q(x))]

Solution

(1) • x : P(x) means, There exists an x such that P(x) holds true and x : P(x) means, “For all x, it is the case that P(x) holds.”

This is true cause if it satisfies for all values of x , it does satisfy for some specific value of x too.

(2)Suppose we claimed, “For every real number, there’s a real number larger than it.” We’d write this as x y : y > x.
The difference between a statement that says x y and a statement that says x y is something to watch out for. For example, if we’re talking about real numbers, then our earlier example x y : y > x is true. writing it with the quantifiers in the other order would become false: y x : y > x. This require a single number that’s greater than every number.

(3)