The contrapositive of statement ` ~ p Rightarrow ( p ^^ ~ q)` is

Options

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

Answer

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

### Step 1: Identify the original statement
The original statement is:
\[ \sim p \Rightarrow (p \land \sim q) \]

### Step 2: Write the contrapositive
The contrapositive of a statement of the form \( A \Rightarrow B \) is \( \sim B \Rightarrow \sim A \). Here, \( A \) is \( \sim p \) and \( B \) is \( p \land \sim q \).

Thus, we need to find:
\[ \sim (p \land \sim q) \Rightarrow \sim (\sim p) \]

### Step 3: Simplify \( \sim (p \land \sim q) \)
Using De Morgan's laws, we can simplify \( \sim (p \land \sim q) \):
\[ \sim (p \land \sim q) = \sim p \lor \sim (\sim q) = \sim p \lor q \]

### Step 4: Simplify \( \sim (\sim p) \)
The negation of \( \sim p \) is:
\[ \sim (\sim p) = p \]

### Step 5: Combine the results
Now we can write the contrapositive:
\[ \sim p \lor q \Rightarrow p \]

### Final Result
Thus, the contrapositive of the statement \( \sim p \Rightarrow (p \land \sim q) \) is:
\[ \sim p \lor q \Rightarrow p \]