Which of the following statements is a tautology

Options

A. ` pvv(p^^q)`
B. ` (p ^^ (prarrq))rarr~q`
C. ` (p^^q)rarr(~(p)rarrq)`
D. `prarr(p^^(rarrq))`

Answer

To determine which of the given statements is a tautology, we need to analyze each statement logically. A tautology is a statement that is always true regardless of the truth values of its components.

### Step-by-Step Solution:

1. **Identify the Statements**:
Let's denote the statements as follows:
- Statement 1: \( P \cup (P \cap Q) \)
- Statement 2: \( (P \cap \neg Q) \cup (P \cap Q) \)
- Statement 3: \( (P \cap Q) \implies P \)
- Statement 4: \( P \implies (Q \implies P) \)

2. **Analyze Statement 1**:
- The expression \( P \cup (P \cap Q) \) can be simplified using the absorption law:
\[ P \cup (P \cap Q) = P \]
- This is not a tautology as it can be false if \( P \) is false.

3. **Analyze Statement 2**:
- The expression \( (P \cap \neg Q) \cup (P \cap Q) \) can be simplified:
\[ (P \cap \neg Q) \cup (P \cap Q) = P \cap (\neg Q \cup Q) = P \]
- This is also not a tautology as it can be false if \( P \) is false.

4. **Analyze Statement 3**:
- The expression \( (P \cap Q) \implies P \) is a logical implication.
- This is always true because if \( P \cap Q \) is true, then \( P \) must be true. Hence, this statement is a tautology.

5. **Analyze Statement 4**:
- The expression \( P \implies (Q \implies P) \) can be analyzed as follows:
- If \( P \) is true, then \( Q \implies P \) is true regardless of \( Q \).
- If \( P \) is false, then the implication \( P \implies (Q \implies P) \) is true since a false premise leads to a true implication.
- Therefore, this statement is also a tautology.

6. **Conclusion**:
- From the analysis, Statements 3 and 4 are tautologies. However, if we are to select just one as the answer based on the given options, Statement 3 is a clear example of a tautology.

### Final Answer:
The statement that is a tautology is **Statement 3: \( (P \cap Q) \implies P \)**.