To determine which of the given statements is true, we need to analyze each statement using truth tables. We will consider the variables \( p \) and \( q \) and their negations.
### Step-by-Step Solution:
1. **Identify Variables and Their Negations**:
- Let \( p \) and \( q \) be the variables.
- The negations are \( \neg p \) and \( \neg q \).
2. **Construct the Truth Table**:
- Since there are two variables, the total number of combinations is \( 2^2 = 4 \).
- The truth table will have the following columns: \( p \), \( q \), \( \neg p \), \( \neg q \).
\[
\begin{array}{|c|c|c|c|}
\hline
p & q & \neg p & \neg q \\
\hline
T & T & F & F \\
T & F & F & T \\
F & T & T & F \\
F & F & T & T \\
\hline
\end{array}
\]
3. **Evaluate Each Statement**:
- **Statement 1**: \( p \iff \neg q \)
- This is true when both values are equal.
- Evaluating:
- \( T \iff F \) → F
- \( T \iff T \) → T
- \( F \iff F \) → T
- \( F \iff T \) → F
- Result: \( F, T, T, F \) → Not a tautology (not all true).
- **Negation of Statement 1**: \( \neg (p \iff \neg q) \)
- This is the opposite of the previous results:
- \( T, F, F, T \) → Not a fallacy (not all false).
- **Statement 2**: \( p \iff q \)
- Evaluating:
- \( T \iff T \) → T
- \( T \iff F \) → F
- \( F \iff T \) → F
- \( F \iff F \) → T
- Result: \( T, F, F, T \) → Not a tautology (not all true).
- **Statement 3**: \( p \land \neg q \)
- This is true when both are true:
- \( T \land F \) → F
- \( T \land T \) → T
- \( F \land F \) → F
- \( F \land T \) → F
- Result: \( F, T, F, F \) → Not a fallacy (not all false).
- **Statement 4**: \( \neg p \land q \)
- Evaluating:
- \( F \land T \) → F
- \( F \land F \) → F
- \( T \land T \) → T
- \( T \land F \) → F
- Result: \( F, F, T, F \) → Not a tautology (not all true).
4. **Conclusion**:
- After evaluating all statements, we find that Statement 2 \( (p \iff q) \) is the only one that is true in some cases but not a tautology or a fallacy.
### Final Answer:
The true statement among the options is **Statement 2: \( p \iff q \)**.
