The statement `~(phArr q)` is not equivalent to

Options

A. `~phArr q`
B. `(p^^ ~q) vv (q ^^ ~q)`
C. `(p vv q) ^^ (~p vv ~q)`
D. `(p vv q) rArr (p ^^ q)`

Answer

To determine which of the given options is not equivalent to the statement \( \neg (p \iff q) \), we will analyze each option step by step.

### Step 1: Understand the original statement
The statement \( \neg (p \iff q) \) means "it is not the case that \( p \) is equivalent to \( q \)." The biconditional \( p \iff q \) is true when both \( p \) and \( q \) are either true or false. Therefore, \( \neg (p \iff q) \) is true when \( p \) and \( q \) have different truth values (i.e., one is true and the other is false).

### Step 2: Analyze each option

#### Option A: \( \neg p \rightarrow q \)
1. The implication \( \neg p \rightarrow q \) can be rewritten using the truth table or logical equivalences.
2. This is equivalent to \( p \lor q \) (using the implication equivalence).
3. The truth values for \( \neg p \rightarrow q \) will yield the same results as \( \neg (p \iff q) \) when evaluated.

#### Option B: \( \neg p \lor q \)
1. This statement means "either \( p \) is false or \( q \) is true."
2. This can also be shown to be equivalent to \( \neg (p \iff q) \) through logical equivalences.
3. Thus, \( \neg p \lor q \) is equivalent to \( \neg (p \iff q) \).

#### Option C: \( p \lor q \land \neg p \lor \neg q \)
1. This statement can be simplified using distribution and logical identities.
2. It is equivalent to \( (p \lor q) \land (\neg p \lor \neg q) \), which simplifies to \( \neg (p \iff q) \).
3. Therefore, this option is also equivalent to \( \neg (p \iff q) \).

#### Option D: \( \neg p \iff q \)
1. This statement means "not \( p \) is equivalent to \( q \)."
2. This is not equivalent to \( \neg (p \iff q) \) because it does not capture the scenario where \( p \) and \( q \) differ.
3. Thus, this option is not equivalent to \( \neg (p \iff q) \).

### Conclusion
After analyzing all options, we find that options A, B, and C are equivalent to \( \neg (p \iff q) \), while option D is not. Therefore, the answer is:

**Option D is not equivalent to \( \neg (p \iff q) \).**

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