The negation of inverse of `~ptoq` is

Options

A. `~qto""p`
B. `p^^~q`
C. `~p^^q`
D. `p^^q`

Answer

To find the negation of the inverse of the statement \( p \to q \), we will follow these steps:

### Step 1: Identify the Inverse
The inverse of a conditional statement \( p \to q \) is given by negating both the hypothesis and the conclusion. Thus, the inverse of \( p \to q \) is:
\[ \sim p \to \sim q \]

### Step 2: Negate the Inverse
Next, we need to find the negation of the inverse we just found. The negation of \( \sim p \to \sim q \) can be expressed using the logical equivalence:
\[ \sim (A \to B) \equiv A \land \sim B \]
Applying this to our case:
\[ \sim (\sim p \to \sim q) \equiv \sim p \land \sim(\sim q) \]
Since \( \sim(\sim q) \) simplifies to \( q \), we have:
\[ \sim (\sim p \to \sim q) \equiv \sim p \land q \]

### Final Result
Thus, the negation of the inverse of \( p \to q \) is:
\[ \sim p \land q \]