The statement `(p to ~q) hArr (p ^^q)` is a-

Options

A. Tautology
B. contradiction
C. Neither tautology nor contradiction
D. None

Answer

To determine whether the statement \( (p \to \neg q) \leftrightarrow (p \land q) \) is a tautology, contradiction, or neither, we can construct a truth table. Here are the steps to solve the problem:

### Step 1: Define the Variables
We have two variables, \( p \) and \( q \). Each can either be true (T) or false (F).

### Step 2: Create the Truth Table
We will create a truth table with columns for \( p \), \( q \), \( \neg q \), \( p \to \neg q \), \( p \land q \), and finally \( (p \to \neg q) \leftrightarrow (p \land q) \).

| \( p \) | \( q \) | \( \neg q \) | \( p \to \neg q \) | \( p \land q \) | \( (p \to \neg q) \leftrightarrow (p \land q) \) |
|---------|---------|---------------|---------------------|------------------|--------------------------------------------------|
| T | T | F | F | T | F |
| T | F | T | T | F | F |
| F | T | F | T | F | F |
| F | F | T | T | F | F |

### Step 3: Fill in the Truth Table
1. **Negation of \( q \)**:
- If \( q \) is T, then \( \neg q \) is F.
- If \( q \) is F, then \( \neg q \) is T.

2. **Implication \( p \to \neg q \)**:
- This is true unless \( p \) is true and \( \neg q \) is false.
- Thus:
- For \( (T, T) \): \( T \to F \) is F.
- For \( (T, F) \): \( T \to T \) is T.
- For \( (F, T) \): \( F \to F \) is T.
- For \( (F, F) \): \( F \to T \) is T.

3. **Conjunction \( p \land q \)**:
- This is true only if both \( p \) and \( q \) are true.
- Thus:
- For \( (T, T) \): T.
- For \( (T, F) \): F.
- For \( (F, T) \): F.
- For \( (F, F) \): F.

4. **Biconditional \( (p \to \neg q) \leftrightarrow (p \land q) \)**:
- This is true if both sides are the same (both true or both false).
- Thus:
- For \( (T, T) \): F.
- For \( (T, F) \): F.
- For \( (F, T) \): F.
- For \( (F, F) \): F.

### Step 4: Analyze the Result
From the truth table, we see that the final column for \( (p \to \neg q) \leftrightarrow (p \land q) \) is all F (false). This means that the statement is a contradiction.

### Conclusion
The statement \( (p \to \neg q) \leftrightarrow (p \land q) \) is a contradiction.

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