The negative of the statement `~p vv (p ^^ q)` is

Options

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

Answer

To find the negation of the statement \( \sim p \lor (p \land q) \), we will follow these steps:

### Step 1: Understand the statement
The given statement is \( \sim p \lor (p \land q) \). Here, \( \sim \) represents negation, \( \lor \) represents disjunction (logical OR), and \( \land \) represents conjunction (logical AND).

### Step 2: Apply De Morgan's Law
To negate the entire statement, we apply De Morgan's Law, which states that the negation of a disjunction is the conjunction of the negations. Thus, we have:
\[ \sim (\sim p \lor (p \land q)) = \sim (\sim p) \land \sim (p \land q) \]

### Step 3: Simplify the negation of each part
1. The negation of \( \sim p \) is simply \( p \).
2. For \( \sim (p \land q) \), we again apply De Morgan's Law:
\[ \sim (p \land q) = \sim p \lor \sim q \]

### Step 4: Combine the results
Now, substituting back into our expression, we have:
\[ p \land (\sim p \lor \sim q) \]

### Step 5: Distribute the conjunction
We can distribute \( p \) over the disjunction:
\[ p \land \sim p \lor p \land \sim q \]

### Step 6: Simplify the expression
The term \( p \land \sim p \) is always false (contradiction), so we can simplify:
\[ \text{False} \lor (p \land \sim q) = p \land \sim q \]

### Final Answer
Thus, the negation of the statement \( \sim p \lor (p \land q) \) is:
\[ p \land \sim q \]

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