Which of the following is logically equivalent to `~(~p implies q)`

Options

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

Answer

To determine which of the given options is logically equivalent to `~(~p implies q)`, we can follow these steps:

### Step 1: Understand the Implication
The expression `~p implies q` can be rewritten using the definition of implication. Recall that `p implies q` is logically equivalent to `~p or q`. Therefore, we can express `~p implies q` as:
\[ ~p \implies q \equiv \neg(~p) \lor q \equiv p \lor q \]

### Step 2: Negate the Implication
Now we need to negate the entire implication:
\[ \sim(~p \implies q) \equiv \sim(p \lor q) \]

### Step 3: Apply De Morgan's Law
Using De Morgan's laws, we can further simplify `~(p or q)`:
\[ \sim(p \lor q) \equiv \sim p \land \sim q \]

### Conclusion
Thus, we find that:
\[ ~(~p \implies q) \equiv \sim p \land \sim q \]

### Final Answer
The expression `~(~p implies q)` is logically equivalent to `~p and ~q`.