Which of the following is logically equivalent to `~(~pto q)`?

Options

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

Explanation

To solve the problem of finding which of the following is logically equivalent to `~(~p → q)`, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Expression**:
The expression we need to simplify is `~(~p → q)`. Here, `~` denotes negation and `→` denotes implication.

2. **Rewrite the Implication**:
Recall that the implication `A → B` can be rewritten as `~A ∨ B`. Therefore, we can rewrite `~p → q` as `~(~p) ∨ q`, which simplifies to `p ∨ q`.

\[ ~p → q \equiv ~(~p) ∨ q \equiv p ∨ q \]

3. **Negate the Expression**:
Now we need to negate the entire expression we just found:
\[ ~(~p → q) \equiv ~(p ∨ q) \]

4. **Apply De Morgan's Law**:
According to De Morgan's laws, the negation of a disjunction is the conjunction of the negations. Thus, we have:
\[ ~(p ∨ q) \equiv ~p ∧ ~q \]

5. **Final Result**:
Therefore, the expression `~(~p → q)` simplifies to `~p ∧ ~q`.

### Conclusion:
The logically equivalent expression to `~(~p → q)` is `~p ∧ ~q`.