If , and are three propositions, then which of the following combination of truth values of , and makes the logical expression false?
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: B
Solution
For an implication A → B to be false, we need A = T and B = F.
With p = F, q = T, r = F:
Antecedent (p ∨ q) ∧ (¬p ∨ r) = (F ∨ T) ∧ (T ∨ F) = T ∧ T = T
Consequent ¬q ∨ r = F ∨ F = F
Since antecedent is T and consequent is F, the implication is F. This is the only combination where the consequent (¬q ∨ r) is false (requiring q = T, r = F) while simultaneously the antecedent remains true (p = F makes ¬p ∨ r = T, and p ∨ q = T since q = T).
With p = F, q = T, r = F:
Antecedent (p ∨ q) ∧ (¬p ∨ r) = (F ∨ T) ∧ (T ∨ F) = T ∧ T = T
Consequent ¬q ∨ r = F ∨ F = F
Since antecedent is T and consequent is F, the implication is F. This is the only combination where the consequent (¬q ∨ r) is false (requiring q = T, r = F) while simultaneously the antecedent remains true (p = F makes ¬p ∨ r = T, and p ∨ q = T since q = T).
[JEE Main 2023 · 29 Jan · Shift 1]