JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)
JEE Main 2023 Mathematics question on Mathematical Reasoning
The compound statement
(∼(P∧Q))∨((∼P)∧Q)⇒((∼P)∧(∼Q)) is equivalent to
- A.
(∼Q)∨P - B.
((∼P)∨Q)∧((∼Q)∨P) - C.
(∼P)∨Q - D.
((∼P)∨Q)∧(∼Q)
Show answer and solution
Correct option: B
Solution
The antecedent simplifies: ~(P∧Q) ∨ (~P∧Q) = (~P∨~Q) ∨ (~P∧Q) = ~P ∨ (~Q∧(P∨1)) = ~P ∨ ~Q (since ~P∨~Q already covers the ~P∧Q case, so the whole antecedent is just ~P∨~Q, i.e., ~(P∧Q)).
The implication ~(P∧Q) ⇒ (~P∧~Q) is equivalent to (P∧Q) ∨ (~P∧~Q), which means P↔Q (both same truth value).
Now check option 2: (~P∨Q) ∧ (~Q∨P) = (P→Q) ∧ (Q→P) = P↔Q.
This matches exactly, so option 2 is equivalent to the given compound statement.
[JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)]
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