For two statements p and q, the statement `~(pvv(~q))` is equivalent to

Options

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

Answer

To solve the problem of determining the equivalence of the statement `~(p ∨ ~q)`, we can use De Morgan's Laws. Here’s a step-by-step breakdown:

### Step 1: Identify the statement
We start with the statement:
\[ S = \sim (p \lor \sim q) \]

### Step 2: Apply De Morgan's Law
According to De Morgan's Law, the negation of a disjunction is equivalent to the conjunction of the negations. This means:
\[ \sim (A \lor B) = \sim A \land \sim B \]
In our case, we can apply this to our statement:
\[ S = \sim p \land \sim (\sim q) \]

### Step 3: Simplify the negation
Next, we simplify the negation of the negation:
\[ \sim (\sim q) = q \]
So, we can rewrite our statement as:
\[ S = \sim p \land q \]

### Conclusion
Thus, the statement `~(p ∨ ~q)` is equivalent to:
\[ \sim p \land q \]