Symbolic Logic Problem and Solution: #2

Symbolic Logic Problem:
We are asked to determine the truth value of statement
S = [P ∙ (Q v X)] v ~[(P ∙ Q) v (P ∙ X)] where X is known to be false and the truth values of P and Q are unknown.

Note: ∙ = the logical "and"
v = the logical "or"
~ = the logical "not"

Solution:

S = [P ∙ (Q v X)] v ~[(P ∙ Q) v (P ∙ X)]

P can be either true or false; Q can be either true or false.

If P is true and Q is true, then (Q v X) is true and [P ∙ (Q v X)] is true. So S is true.

If P is true and Q is false, then (P ∙ Q) is false. (P ∙ X) is always false since X is false.

So [(P ∙ Q) v (P ∙ X)] is false and ~[(P ∙ Q) v (P ∙ X)] is true. So S is true.

If P is false and Q is true, then (P ∙ Q) is false ->> [(P ∙ Q) v (P ∙ X)] is false.

Thus, ~[(P ∙ Q) v (P ∙ X)] is true ->> S is true.

If P is false and Q is false, then (P ∙ Q) is false ->> [(P ∙ Q) v (P ∙ X)] is false.

Thus, ~[(P ∙ Q) v (P ∙ X)] is true ->> S is true.

Thus, S is false always.