Question

The statement \( \neg(p \leftrightarrow q) \) is logically equivalent to:

• (A) \( \neg p \leftrightarrow \neg q \)
• (B) \( \neg p \to q \)
• (C) \( \neg(p \to \neg q) \)
• (D) \( p \leftrightarrow \neg q \)

Hint:

Negating one side of a biconditional is equivalent to negating the entire biconditional: \[ \neg(p\leftrightarrow q)\equiv(p\leftrightarrow\neg q). \] This is a very important logical identity.

Answer 1

The Correct Option is D

Concept:
The biconditional statement \[ p\leftrightarrow q \] is true whenever both propositions have the same truth values. Its negation, \[ \neg(p\leftrightarrow q), \] becomes true whenever the truth values are different.

Step 1: Understanding the biconditional statement.
The statement \[ p\leftrightarrow q \] means: \[ (p\to q)\wedge(q\to p). \] This statement is true in the following cases: \[ (T,T)\quad \text{and}\quad(F,F). \] Thus, it is false when: \[ (T,F)\quad \text{or}\quad(F,T). \]

Step 2: Negating the biconditional statement.
Therefore, \[ \neg(p\leftrightarrow q) \] is true only when \( p \) and \( q \) have opposite truth values. So this represents the XOR operation.

Step 3: Checking option (D).
Consider: \[ p\leftrightarrow \neg q. \] This statement becomes true when: \[ p \text{ and } \neg q \] have the same truth values. This implies: \[ p \text{ and } q \] must have opposite truth values. Hence, \[ p\leftrightarrow\neg q \] has exactly the same truth table as \[ \neg(p\leftrightarrow q). \] Therefore, \[ \boxed{\neg(p\leftrightarrow q)\equiv p\leftrightarrow\neg q}. \]