Symbolic Logic Problem and Solution: #1

We are asked to determine the truth value of statement
S = [P v (Q v X)] ∙ ~[(P v Q) v 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 v (Q v X)] ∙ ~[(P v Q) v X]

P is either true or false; Q is either true or false.

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

(P v Q) is true ->> [(P v Q) v X] is true ->> ~[(P v Q) v X] is false ->> S is false.

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

~[(P v Q) v X] is false ->> S is false.

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

~[(P v Q) v X] is false ->> S is false.

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

~[(P v Q) v X] is true. But since P is false and Q and X are each false, (Q v X) is false and [P v (Q v X)] is false ->> S is false.

Thus, S is false always.