The properties `(p to ~q)^^(~ p to p) ` is a

Options

A. tautology and contradiction
B. neither tautology nor contradiction
C. contradiction
D. tautology

Explanation

To determine whether the property \( (p \to \neg q) \land (\neg p \to p) \) is a tautology, contradiction, or neither, we can follow these steps:

### Step 1: Define the expressions
We have two implications:
1. \( A = p \to \neg q \)
2. \( B = \neg p \to p \)

### Step 2: Construct the truth table
We will create a truth table to evaluate the truth values of \( A \) and \( B \) for all combinations of truth values of \( p \) and \( q \).

| \( p \) | \( q \) | \( \neg q \) | \( A = p \to \neg q \) | \( \neg p \) | \( B = \neg p \to p \) | \( A \land B \) |
|---------|---------|---------------|-------------------------|--------------|------------------------|------------------|
| T | T | F | F | F | T | F |
| T | F | T | T | F | T | T |
| F | T | F | T | T | F | F |
| F | F | T | T | T | F | F |

### Step 3: Evaluate \( A \) and \( B \)
- For \( A = p \to \neg q \):
- When \( p = T \) and \( q = T \), \( A = F \)
- When \( p = T \) and \( q = F \), \( A = T \)
- When \( p = F \) and \( q = T \), \( A = T \)
- When \( p = F \) and \( q = F \), \( A = T \)

- For \( B = \neg p \to p \):
- When \( p = T \), \( B = T \)
- When \( p = F \), \( B = F \)

### Step 4: Evaluate \( A \land B \)
Now, we combine the results of \( A \) and \( B \):
- For \( (T, T) \): \( A \land B = F \)
- For \( (T, F) \): \( A \land B = T \)
- For \( (F, T) \): \( A \land B = F \)
- For \( (F, F) \): \( A \land B = F \)

### Step 5: Analyze the final results
The final column \( A \land B \) gives us:
- F
- T
- F
- F

Since the result is not always true (it has both true and false values), we conclude that the expression is neither a tautology nor a contradiction.

### Conclusion
The property \( (p \to \neg q) \land (\neg p \to p) \) is **neither a tautology nor a contradiction**.

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