Step 1: Understanding the Concept:
In mathematical logic, the biconditional statement \( p \leftrightarrow q \) (if and only if) is true only when both \( p \) and \( q \) share the same truth value.
The negation of this statement, \( \neg(p \leftrightarrow q) \), therefore denotes that \( p \) and \( q \) must have different truth values.
This logical structure corresponds to the "Exclusive OR" (XOR) operation, which returns True only if exactly one of the inputs is True.
We can evaluate equivalence by constructing truth tables or by applying established laws of propositional logic.
Step 2: Key Formula or Approach:
The truth table for the base statement \( p \leftrightarrow q \) is:
- \( T \leftrightarrow T = T \)
- \( T \leftrightarrow F = F \)
- \( F \leftrightarrow T = F \)
- \( F \leftrightarrow F = T \)
Negating these results gives:
- \( \neg(T \leftrightarrow T) = F \)
- \( \neg(T \leftrightarrow F) = T \)
- \( \neg(F \leftrightarrow T) = T \)
- \( \neg(F \leftrightarrow F) = F \)
We seek an option that produces this exact sequence of results.
Step 3: Detailed Explanation:
Let's analyze Option (D): \( p \leftrightarrow \neg q \).
This statement asserts that \( p \) is true if and only if \( q \) is false.
Let's check its truth values:
1. If \( p = T, q = T \): then \( \neg q = F \). Statement becomes \( T \leftrightarrow F \), which is \( F \).
2. If \( p = T, q = F \): then \( \neg q = T \). Statement becomes \( T \leftrightarrow T \), which is \( T \).
3. If \( p = F, q = T \): then \( \neg q = F \). Statement becomes \( F \leftrightarrow F \), which is \( T \).
4. If \( p = F, q = F \): then \( \neg q = T \). Statement becomes \( F \leftrightarrow T \), which is \( F \).
The truth values for Option (D) are \( F, T, T, F \).
Comparing these to our negated biconditional truth values (\( F, T, T, F \)), we see they are an exact match.
Alternatively, using logical properties:
\( \neg(p \leftrightarrow q) \equiv (p \land \neg q) \lor (\neg p \land q) \)
\( p \leftrightarrow \neg q \equiv (p \to \neg q) \land (\neg q \to p) \equiv (\neg p \lor \neg q) \land (q \lor p) \equiv (p \land \neg q) \lor (\neg p \land q) \)
The equivalence is proven.
Step 4: Final Answer:
By evaluating the truth tables, we confirmed that the negation of the biconditional requires \( p \) and \( q \) to have opposite truth values. Option (D) expresses this same condition. This corresponds to Option (D).