`~(~p^^q), "is equal to" `

Options

A. p`vv`(~q)
B. p`vv`q
C. p`^^`(~q)
D. ~p`^^`~q

Answer

To solve the expression \( \sim(\sim p \land q) \), we can apply the rules of logical negation and conjunction. Let's break this down step by step.

### Step 1: Understand the expression
We start with the expression \( \sim(\sim p \land q) \). Here, \( \sim \) denotes negation, and \( \land \) denotes logical conjunction (AND).

### Step 2: Apply De Morgan's Law
According to De Morgan's Laws, the negation of a conjunction can be expressed as the disjunction of the negations. Specifically, we have:
\[ \sim(A \land B) = \sim A \lor \sim B \]
In our case, we can apply this to \( \sim(\sim p \land q) \):
\[ \sim(\sim p \land q) = \sim(\sim p) \lor \sim q \]

### Step 3: Simplify the negation
Next, we simplify \( \sim(\sim p) \). The negation of a negation is simply the original statement:
\[ \sim(\sim p) = p \]
So, substituting this back into our expression gives:
\[ \sim(\sim p \land q) = p \lor \sim q \]

### Final Result
Thus, the expression \( \sim(\sim p \land q) \) simplifies to:
\[ p \lor \sim q \]

### Summary
The expression \( \sim(\sim p \land q) \) is equal to \( p \lor \sim q \).