If p is any statement then `(p vee ~p)` is a

Options

A. Contingency
B. Contradiction
C. Tautology
D. None of these

Answer

To determine the nature of the statement \( p \vee \neg p \) (where \( \vee \) represents logical OR and \( \neg \) represents negation), we can use a truth table. Here’s a step-by-step solution:

### Step 1: Define the Variables
Let \( p \) be any statement. The two possible truth values for \( p \) are:
- True (T)
- False (F)

### Step 2: Determine the Negation
The negation of \( p \), denoted as \( \neg p \), will have the following values:
- If \( p \) is True (T), then \( \neg p \) is False (F).
- If \( p \) is False (F), then \( \neg p \) is True (T).

### Step 3: Create the Truth Table
Now we can create the truth table for \( p \) and \( \neg p \):

| \( p \) | \( \neg p \) | \( p \vee \neg p \) |
|---------|---------------|---------------------|
| T | F | T |
| F | T | T |

### Step 4: Analyze the Results
In the truth table, the column for \( p \vee \neg p \) shows that:
- When \( p \) is True, \( p \vee \neg p \) is True.
- When \( p \) is False, \( p \vee \neg p \) is also True.

### Step 5: Conclusion
Since \( p \vee \neg p \) is always True regardless of the truth value of \( p \), we conclude that the statement \( p \vee \neg p \) is a **tautology**.

### Final Answer
Thus, \( p \vee \neg p \) is a tautology.

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