If p and q are two statements then negation of `p hArr (q rArr p)` is

Options

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

Answer

To find the negation of the statement \( p \leftrightarrow (q \rightarrow p) \), we can follow these steps:

### Step 1: Understand the statement
The statement \( p \leftrightarrow (q \rightarrow p) \) means that \( p \) is true if and only if \( q \) implies \( p \).

### Step 2: Rewrite the implication
Recall that \( q \rightarrow p \) can be rewritten using logical equivalences. Specifically, \( q \rightarrow p \) is equivalent to \( \neg q \lor p \). Thus, we can rewrite the original statement as:
\[ p \leftrightarrow (\neg q \lor p) \]

### Step 3: Apply the biconditional
The biconditional \( p \leftrightarrow r \) can be expressed as:
\[ (p \rightarrow r) \land (r \rightarrow p) \]
Applying this to our statement, we have:
\[ (p \rightarrow (\neg q \lor p)) \land ((\neg q \lor p) \rightarrow p) \]

### Step 4: Simplify the implications
1. For \( p \rightarrow (\neg q \lor p) \):
- This is always true because if \( p \) is true, \( \neg q \lor p \) is true regardless of \( q \).

2. For \( (\neg q \lor p) \rightarrow p \):
- This is equivalent to \( \neg(\neg q \lor p) \lor p \), which simplifies to \( (q \land \neg p) \lor p \). This is also always true because if \( p \) is true, the whole expression is true.

### Step 5: Combine the results
Since both parts of the biconditional are always true, we conclude that:
\[ p \leftrightarrow (\neg q \lor p) \text{ is a tautology.} \]

### Step 6: Negate the statement
Now, we need to find the negation of the entire statement:
\[ \neg(p \leftrightarrow (\neg q \lor p)) \]
Since the original statement is a tautology, its negation is always false:
\[ \neg(p \leftrightarrow (\neg q \lor p)) \text{ is always false.} \]

### Final Answer
Thus, the negation of \( p \leftrightarrow (q \rightarrow p) \) is:
\[ \text{False} \]