The statemetn `p to ( p vvq)` is a-

Options

A. Tautology
B. Contiadictin
C. Neither tautology nor contradiction
D. None of these

Answer

To determine whether the statement \( p \to (p \lor q) \) is a tautology, contradiction, or neither, we can use a truth table. Let's break down the solution step by step.

### Step 1: Define the Statement
The statement we are analyzing is \( p \to (p \lor q) \).

### Step 2: Create the Truth Table
We need to evaluate the truth values of \( p \), \( q \), \( p \lor q \), and \( p \to (p \lor q) \).

1. **List the possible truth values for \( p \) and \( q \)**:
- \( p \): True (T), True (T), False (F), False (F)
- \( q \): True (T), False (F), True (T), False (F)

2. **Calculate \( p \lor q \)**:
- \( p \lor q \) is true if at least one of \( p \) or \( q \) is true.
- The results for \( p \lor q \) will be:
- T (T or T)
- T (T or F)
- T (F or T)
- F (F or F)

3. **Calculate \( p \to (p \lor q) \)**:
- The implication \( p \to (p \lor q) \) is false only when \( p \) is true and \( (p \lor q) \) is false.
- The results for \( p \to (p \lor q) \) will be:
- T (T → T)
- T (T → T)
- T (F → T)
- T (F → F)

### Step 3: Fill in the Truth Table
| \( p \) | \( q \) | \( p \lor q \) | \( p \to (p \lor q) \) |
|---------|---------|----------------|-------------------------|
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | T |

### Step 4: Analyze the Results
From the truth table, we see that the column for \( p \to (p \lor q) \) is always true (T) regardless of the truth values of \( p \) and \( q \).

### Conclusion
Since \( p \to (p \lor q) \) is true for all possible truth values of \( p \) and \( q \), we conclude that the statement is a **tautology**.

### Final Answer
The statement \( p \to (p \lor q) \) is a tautology.

---