Question

The number of ordered triplets of the truth values of \( p, q, r \) and such that the truth value of the statement \[ (p \lor q) \land (p \lor r) \implies (q \lor r) \text{ is True, is equal to:} \]

Hint:

Constructing truth tables is an effective way to check the validity of logical statements by analyzing all possible truth values.

Answer 1

Correct Answer: 7

(p ∨ q) ∧ (p ∨ r) ⇒ (q ∨ r) is true if either (p ∨ q) ∧ (p ∨ r) is false or (q ∨ r) is true. We analyze all possible truth values of p, q, r to find the number of ordered triplets: 

• Case 1: (p ∨ q) ∧ (p ∨ r) is false. This occurs when both p ∨ q and p ∨ r are false, implying p = q = r = false. (q ∨ r) is false too, so the implication is true.
  Triplet: (false, false, false).
• Case 2: (q ∨ r) is true. Possible truth assignments are:
  • q = true, r = false: (p ∨ q) ∧ (p ∨ r) is true if p = true or q = true.
  • q = false, r = true: (p ∨ q) ∧ (p ∨ r) is true if p = true or r = true.
  • q = true, r = true: Every value of p makes the conjunction true.

Total true triplets:

(0,0,0)		(0,1,0)	(1,1,0)
(0,0,1)	(1,0,1)	(0,1,1)	(1,1,1)

Seven triplets satisfy the condition.