MHT CET · Maths · Mathematical Reasoning

Negation of the statement ' Horses have wings if and only if crows have tails. ' is

A Horses have wings but crows do not have tails.
or crows have tails but horses do not have wings.
B Horses do not have wings if and only if crows do not have tails.
C Horses do not have wings and crows have tails.
or Horses have wings and crows have tails.
D Horses do not have wings and crows do not have tails.

Verified Solution

Answer & Solution

Correct Answer

(A) Horses have wings but crows do not have tails.
or crows have tails but horses do not have wings.

Step-by-step Solution

Detailed explanation

Let p : Horses have wings
q : Crows have tails
Given statement is \(\mathrm{p} \leftrightarrow \mathrm{q}\)
\(\sim(p \leftrightarrow q) \equiv(p \wedge \sim q) \vee(q \wedge \sim p)\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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