The statement ` p implies ~ (p ^^ ~ q)` is -

Options

A. contradiction
B. a tautology
C. either (A) or (B)
D. neither (A) nor (B)

Answer

To determine whether the statement \( p \implies \neg (p \land \neg q) \) is a contradiction, a tautology, or neither, we can use a truth table. Here’s how to solve it step by step:

### Step 1: Identify the components of the statement
The statement consists of:
- \( p \)
- \( q \)
- \( \neg q \) (negation of \( q \))
- \( p \land \neg q \) (conjunction of \( p \) and \( \neg q \))
- \( \neg (p \land \neg q) \) (negation of the conjunction)
- \( p \implies \neg (p \land \neg q) \) (implication)

### Step 2: Create a truth table
We will create a truth table to evaluate the statement. The truth values for \( p \) and \( q \) can be either True (T) or False (F).

| \( p \) | \( q \) | \( \neg q \) | \( p \land \neg q \) | \( \neg (p \land \neg q) \) | \( p \implies \neg (p \land \neg q) \) |
|---------|---------|---------------|-----------------------|------------------------------|---------------------------------------|
| T | T | F | F | T | T |
| T | F | T | T | F | F |
| F | T | F | F | T | T |
| F | F | T | F | T | T |

### Step 3: Analyze the truth table
Now, we will analyze the last column of the truth table, which represents the value of the statement \( p \implies \neg (p \land \neg q) \).

- The values are: T, F, T, T.

### Step 4: Determine the nature of the statement
- A **tautology** is a statement that is always true (all T).
- A **contradiction** is a statement that is always false (all F).
- Since the statement has both true and false values, it is **neither** a tautology nor a contradiction.

### Conclusion
Thus, the statement \( p \implies \neg (p \land \neg q) \) is neither a tautology nor a contradiction.