Netgation of `(pRightarrowq)Rightarrow(qRightarrowp)` is

Options

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

Answer

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

### Step 1: Understand the implications
Recall that the implication \(p \Rightarrow q\) can be rewritten in terms of logical operators:
\[ p \Rightarrow q \equiv \neg p \lor q \]
Thus, we can rewrite the entire expression:
\[ (p \Rightarrow q) \Rightarrow (q \Rightarrow p) \equiv (\neg p \lor q) \Rightarrow (\neg q \lor p) \]

### Step 2: Rewrite the outer implication
Using the same logic for the outer implication:
\[ A \Rightarrow B \equiv \neg A \lor B \]
We can rewrite our expression as:
\[ \neg(\neg p \lor q) \lor (\neg q \lor p) \]

### Step 3: Apply De Morgan's Law
Now we apply De Morgan's Law to \(\neg(\neg p \lor q)\):
\[ \neg(\neg p \lor q) \equiv p \land \neg q \]
Thus, we can rewrite our expression as:
\[ (p \land \neg q) \lor (\neg q \lor p) \]

### Step 4: Simplify the expression
Now we can simplify the expression:
\[ (p \land \neg q) \lor (\neg q \lor p) \equiv \neg q \lor p \]
This is because if \(p\) is true, the expression holds true regardless of \(q\), and if \(q\) is false, the expression also holds true.

### Step 5: Negate the entire expression
Now we need to find the negation of the entire expression:
\[ \neg((p \Rightarrow q) \Rightarrow (q \Rightarrow p)) \equiv \neg(\neg q \lor p) \]
Using De Morgan's Law again:
\[ \neg(\neg q \lor p) \equiv q \land \neg p \]

### Final Result
Thus, the negation of the statement \((p \Rightarrow q) \Rightarrow (q \Rightarrow p)\) is:
\[ q \land \neg p \]