The question is asking to determine the logical equivalence of $\sim (A \vee B)$ using the provided options. Let me solve it step by step:

Step 1: Apply De Morgan's Laws

The statement $\sim (A \vee B)$ can be rewritten using De Morgan's Laws, which state: $\sim (A \vee B) = (\sim A \wedge \sim B)$

Step 2: Compare to the Options

Now we examine the given options:

• Option a: $A \vee B$
  This is simply the disjunction of $A$ and $B$. Not equivalent.

• Option b: $\sim A \vee \sim B$
  This is the negation of the conjunction ($A \wedge B$), but it does not match $\sim (A \vee B)$. Not equivalent.

• Option c: $\sim A \wedge \sim B$
  This matches $\sim (A \vee B)$ exactly, based on De Morgan's Laws. Correct.

• Option d: $\sim A \wedge B$
  This is a different conjunction and does not represent $\sim (A \vee B)$. Not equivalent.

Final Answer:

Option c: $\sim A \wedge \sim B$ is the correct choice.

Would you like a detailed explanation of De Morgan's Laws or help with another concept?

5 Related Questions:

1. What are De Morgan's Laws and how are they applied in logic?
2. How can logical equivalences simplify expressions?
3. What is the difference between a conjunction ($\wedge$) and disjunction ($\vee$)?
4. How would the equivalence change for $\sim (A \wedge B)$?
5. Can you provide truth tables to confirm logical equivalence?

Tip:

Always use truth tables to verify logical equivalences if you're unsure about the results!