Solution

The correct answer is Option 3

Key Points

• Statement (a): "If she is intelligent, then she is unhappy."
  This is represented as P → ¬Q, meaning if "P" (she is intelligent) is true, then "¬Q" (she is unhappy) must be true.
• Statement (b): "She is neither intelligent nor happy."
  This is represented as ¬P ∧ ¬Q, meaning both "¬P" (she is not intelligent) and "¬Q" (she is not happy) must be true.
• Statement (c): "It is necessary to be not intelligent in order to be happy."
  This is represented as ¬Q → ¬P, meaning if "¬Q" (she is unhappy) is true, then "¬P" (she is not intelligent) must also be true.
• Statement (d): "To be not intelligent is to be unhappy."
  This is represented as ¬P → ¬Q, meaning if "¬P" (she is not intelligent) is true, then "¬Q" (she is unhappy) must also be true.

Additional Information

• The propositional expressions for each statement are derived based on logical equivalence and their meanings.
• Option 3 correctly represents the propositional expressions for the given statements:
  • (a) P → ¬Q
  • (b) ¬P ∧ ¬Q
  • (c) ¬Q → ¬P
  • (d) ¬P → ¬Q

Hence, the correct answer is: Option 3