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 will analyze it step by step using a truth table.

### Step 1: Identify the components
We have two propositions: \( p \) and \( q \). The statement we are analyzing is \( p \to (p \lor q) \).

### Step 2: Create a truth table
We will create a truth table with the following columns:
1. \( p \)
2. \( q \)
3. \( p \lor q \) (the disjunction of \( p \) and \( q \))
4. \( p \to (p \lor q) \) (the implication we are analyzing)

### Step 3: Fill in the truth values for \( p \) and \( q \)
We will consider all 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)

### Step 4: Calculate \( p \lor q \)
Now we will determine the values for \( p \lor q \):
- When \( p \) is T and \( q \) is T: \( p \lor q \) is T
- When \( p \) is T and \( q \) is F: \( p \lor q \) is T
- When \( p \) is F and \( q \) is T: \( p \lor q \) is T
- When \( p \) is F and \( q \) is F: \( p \lor q \) is F

### Step 5: Calculate \( p \to (p \lor q) \)
Now we will determine the values for \( p \to (p \lor q) \):
- When \( p \) is T and \( p \lor q \) is T: \( p \to (p \lor q) \) is T
- When \( p \) is T and \( p \lor q \) is T: \( p \to (p \lor q) \) is T
- When \( p \) is F and \( p \lor q \) is T: \( p \to (p \lor q) \) is T
- When \( p \) is F and \( p \lor q \) is F: \( p \to (p \lor q) \) is T

### Step 6: Compile the truth table
Now we can compile 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 7: Analyze the results
From the truth table, we see that the last column \( p \to (p \lor q) \) is always true regardless of the truth values of \( p \) and \( q \). Therefore, the statement \( p \to (p \lor q) \) is a tautology.

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

---