What if R is false in your ist point ?

1 vote

57. Let P, Q, R and S be Propositions. Assume that the equivalences P ⇔ (Q ∨ ¬ Q) and Q ⇔ R hold. Then the truth value of the formula (P ∧ Q) ⇒ ((P ∧ R) ∨ S) is always :

(1) True

(2) False

(3) Same as truth table of Q

(4) Same as truth table of S

(1) True

(2) False

(3) Same as truth table of Q

(4) Same as truth table of S

5 votes

Best answer

P ⇔ (Q ∨ ¬ Q) "**P should be true because RHS will be TRUE always **"

Q ⇔ R "**when Q is true R is true" and "when Q is false R is false**"

$(P ∧ Q) ⇒ ((P ∧ R) ∨ S)$

there can be only 2 cases (value of S doesn't matter)

1) P = True, Q = True and R = True

$(T ∧ T) ⇒ ((T ∧ T) ∨ S)$

so this case is True

2) P =True, Q = R = False

$(T ∧ F) ⇒ ((P ∧ R) ∨ S)$

In case implication if the premises is false then whole statement is true

this case is also true

The given expression is True in both the cases.

Answer is 1) True