Question

‘When it is raining, peacocks dance.’
Based only on this sentence, which one of the following options is necessarily
true?

• A. Peacocks dance only when it is raining.
• B. When peacocks dance, it is raining.
• C. When peacocks are not dancing, it is not raining.
• D. When it is not raining, peacocks do not dance.

Hint:

Remember that a statement if R then D is logically equivalent only to its contrapositive if not D then not R, not to its converse or inverse.

Answer 1

The Correct Option is C

Step 1: The given statement is a conditional: if it is raining, then peacocks dance. In logical form, let R denote it is raining and D denote peacocks dance. The statement is \(R \rightarrow D\).
Step 2: A conditional statement \(R \rightarrow D\) is logically equivalent only to its contrapositive, which is \(\neg D \rightarrow \neg R\), meaning if peacocks are not dancing, then it is not raining.
Step 3: Check each option against this. Option (A), peacocks dance only when it is raining, actually states \(D \rightarrow R\), which is the converse, not guaranteed by the original statement. Option (B), when peacocks dance it is raining, is also the converse \(D \rightarrow R\), not necessarily true. Option (D), when it is not raining peacocks do not dance, states \(\neg R \rightarrow \neg D\), which is the inverse, also not necessarily true.
Step 4: Option (C), when peacocks are not dancing it is not raining, states exactly \(\neg D \rightarrow \neg R\), the contrapositive of the original statement, which is always logically true whenever the original statement is true.
Final Answer: Option (C).