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

Options

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

Answer

To solve the question, we need to find the logical equivalent of the expression `~(p ∨ q)`, where `~` denotes negation and `∨` denotes disjunction (logical OR). We will use De Morgan's laws to transform the expression.

### Step-by-Step Solution:

1. **Identify the Expression**: We start with the expression `~(p ∨ q)`.

2. **Apply De Morgan's Law**: According to De Morgan's laws, the negation of a disjunction can be expressed as the conjunction of the negations. Specifically, De Morgan's law states that:
\[ \sim (p \lor q) \equiv \sim p \land \sim q \]
This means that the negation of `p ∨ q` is equivalent to `¬p ∧ ¬q`.

3. **Write the Equivalent Expression**: Therefore, we can rewrite `~(p ∨ q)` as:
\[ \sim p \land \sim q \]

4. **Conclusion**: The expression `~(p ∨ q)` is logically equivalent to `¬p ∧ ¬q`.

### Final Answer:
The answer is `¬p ∧ ¬q`.