Question

Question:

Which of the following statements is logically equivalent to:
"If it rains, I will take an umbrella"?

Statements:
A. If I take an umbrella, it will rain.
B. If I do not take an umbrella, it will not rain.
C. If it does not rain, I will not take an umbrella.
D. If I do not take an umbrella, it will rain.

Answer Options:
(1) A and B
(2) C only
(3) B and C
(4) D only

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Solution:

Original Statement Analysis:

The given statement "If it rains, I will take an umbrella" can be represented logically as:

• P → Q ("If P, then Q"), where:
  • P = It rains.
  • Q = I will take an umbrella.

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Logical Equivalences:

1. Contrapositive:
   The contrapositive of "If P → Q" is logically equivalent:

   • ~Q → ~P ("If I do not take an umbrella, it will not rain").

2. Inverse and Converse:
   These are not necessarily equivalent to the original statement.

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Evaluating the Options:

1. Statement A:
   "If I take an umbrella, it will rain."

   • This is the converse of the original statement (Q → P), which is not logically equivalent to P → Q.
   • Incorrect.

2. Statement B:
   "If I do not take an umbrella, it will not rain."

   • This is the contrapositive of the original statement (~Q → ~P).
   • Correct.

3. Statement C:
   "If it does not rain, I will not take an umbrella."

   • This directly follows from the original statement, as taking an umbrella depends on rain.
   • Correct.

4. Statement D:
   "If I do not take an umbrella, it will rain."

   • This is not logically equivalent, as it suggests the reverse causation.
   • Incorrect.

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Answer: (3) B and C