To determine which of the given options is a tautology, we will analyze each option using truth tables. A tautology is a statement that is true in every possible interpretation.
### Step-by-Step Solution:
**Step 1: Analyze Option A**
1. **Construct the truth table for Option A:**
- Let \( p \) and \( q \) take the values T (True) and F (False).
- The combinations of \( p \) and \( q \) are:
- \( p = T, q = T \)
- \( p = T, q = F \)
- \( p = F, q = T \)
- \( p = F, q = F \)
2. **Calculate \( p \to q \):**
- \( T \to T = T \)
- \( T \to F = F \)
- \( F \to T = T \)
- \( F \to F = T \)
Result: \( p \to q = T, F, T, T \)
3. **Calculate \( p \land (p \to q) \):**
- \( T \land T = T \)
- \( T \land F = F \)
- \( F \land T = F \)
- \( F \land T = F \)
Result: \( p \land (p \to q) = T, F, F, F \)
4. **Calculate \( (p \land (p \to q)) \lor q \):**
- \( T \lor T = T \)
- \( F \lor F = F \)
- \( F \lor T = T \)
- \( F \lor F = F \)
Result: \( (p \land (p \to q)) \lor q = T, F, T, F \)
Since the final result is not always true, Option A is not a tautology.
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**Step 2: Analyze Option B**
1. **Construct the truth table for Option B:**
- Same combinations of \( p \) and \( q \).
2. **Calculate \( p \to q \):**
- Same results as before: \( T, F, T, T \)
3. **Calculate \( p \land (p \to q) \):**
- Same results as before: \( T, F, F, F \)
4. **Calculate \( q \to (p \land (p \to q)) \):**
- \( T \to T = T \)
- \( F \to F = T \)
- \( T \to F = F \)
- \( F \to F = T \)
Result: \( q \to (p \land (p \to q)) = T, T, F, T \)
Since the final result is not always true, Option B is not a tautology.
---
**Step 3: Analyze Option C**
1. **Construct the truth table for Option C:**
- Same combinations of \( p \) and \( q \).
2. **Calculate \( p \land q \):**
- \( T \land T = T \)
- \( T \land F = F \)
- \( F \land T = F \)
- \( F \land F = F \)
Result: \( p \land q = T, F, F, F \)
3. **Calculate \( p \lor (p \land q) \):**
- \( T \lor T = T \)
- \( T \lor F = T \)
- \( F \lor F = F \)
- \( F \lor F = F \)
Result: \( p \lor (p \land q) = T, T, F, F \)
Since the final result is not always true, Option C is not a tautology.
---
**Step 4: Analyze Option D**
1. **Construct the truth table for Option D:**
- Same combinations of \( p \) and \( q \).
2. **Calculate \( p \land (p \lor q) \):**
- \( T \land T = T \)
- \( T \land T = T \)
- \( F \land T = F \)
- \( F \land F = F \)
Result: \( p \land (p \lor q) = T, T, F, F \)
Since the final result is not always true, Option D is not a tautology.
---
### Conclusion:
None of the options A, B, C, or D are tautologies based on the truth table analysis.
