Let `(p to q) leftrightarrow (~q**p)` is a tautology , then `p**~q` is equivalent to

Options

A. `(p to q)`
B. `(p vv q)`
C. `(p leftrightarrow q)`
D. `p^^q`

Answer

To solve the problem, we need to analyze the statement given and find the equivalent expression for \( p \land \neg q \) based on the tautology condition provided.

### Step-by-Step Solution:

1. **Understanding the Tautology**:
We are given that \( (p \implies q) \leftrightarrow (\neg q \land p) \) is a tautology. A tautology is a statement that is always true regardless of the truth values of its components.

2. **Truth Table Construction**:
To analyze the expression, we will create a truth table for \( p \) and \( q \) and evaluate both sides of the biconditional.

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

From the truth table, we see that the biconditional is not always true. Therefore, we need to analyze further.

3. **Revising the Biconditional**:
For \( (p \implies q) \leftrightarrow (\neg q \land p) \) to be a tautology, both sides must yield the same truth values under all circumstances. This means that both expressions must be equivalent.

4. **Expressing Implication**:
Recall that \( p \implies q \) can be rewritten as \( \neg p \lor q \). Thus, we can rewrite the biconditional:
\[ (\neg p \lor q) \leftrightarrow (\neg q \land p) \]

5. **Finding the Equivalent Expression**:
We need to find what \( p \land \neg q \) is equivalent to. We know that:
\[ p \land \neg q \text{ is equivalent to } \neg(\neg p \lor q) \]
This means we can express \( p \land \neg q \) in terms of \( p \) and \( q \).

6. **Using De Morgan's Laws**:
Using De Morgan's laws, we can rewrite:
\[ \neg(\neg p \lor q) = p \land \neg q \]
This means that \( p \land \neg q \) is equivalent to the negation of the disjunction of \( \neg p \) and \( q \).

7. **Final Expression**:
Thus, we conclude that:
\[ p \land \neg q \text{ is equivalent to } \neg(\neg p \lor q) \]

### Final Result:
The expression \( p \land \neg q \) is equivalent to \( \neg(\neg p \lor q) \).