Logic , Quantifiers , Logical Formula

Question

Please note that I'll be using the following notations as follows, throughout the question.

VX - For all X (Universal Quantifier)

EX - For some X(Existential quantifier)

Consider the logic formulas G1 and G2 :

G1 : (VX)[ P(X) V Q(X) ] --> (VX)P(X) V (VX)Q(X)

G2 : (EX)P(X) --> (VX)P(X)

Can anyone please explain how do we obtain the intuitive meaning of G1 and G2 under the interpretation I over the set of integers ; under which

P(X) means that X is even and

Q(X) means that X is odd.

Answer 1

Neither G1 nor G2 are true.

Let's use as suggested, that P(X) means that X is even and Q(X) means that X is odd,

G1 now becomes:

For all X, (X is odd or X is even) -> (For all X, X is odd) or (For all X, X is even)

The first part is true, since for all integers X, X is odd or X is even is true.

However, (For all X, X is odd) is false, and also (For all X, X is even) is false, so

[ (For all X, X is odd) OR (For all X, X is even) ] is false,

G2 now becomes:

There exists an integer X such that X is even -> For all integers X, X is even.

Again, the first part is true, since there are many even integers. However, the second part is false, since there are also many odd integers.