The converse of `p rArr (q rArr r)` is

Options

A. `(q^^-r)vvp`
B. `(~qvvr)vvp`
C. `(q^^~r)^^~p`
D. `(q^^~r)^^p`

Answer

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

### Step 1: Understand the original statement
The original statement is \( p \Rightarrow (q \Rightarrow r) \). This means "if \( p \) is true, then \( q \) implies \( r \)".

### Step 2: Write the converse
The converse of a statement \( A \Rightarrow B \) is \( B \Rightarrow A \). In our case, we need to express the converse of \( p \Rightarrow (q \Rightarrow r) \).

So, we will write:
\[ (q \Rightarrow r) \Rightarrow p \]

### Step 3: Simplify the expression
Next, we can simplify the expression \( (q \Rightarrow r) \Rightarrow p \). We know that \( q \Rightarrow r \) can be rewritten using logical equivalence:
\[ q \Rightarrow r \equiv \neg q \lor r \]

Thus, we can rewrite the converse as:
\[ (\neg q \lor r) \Rightarrow p \]

### Step 4: Further simplification
Now, we can rewrite the implication \( (\neg q \lor r) \Rightarrow p \) using the logical equivalence of implications:
\[ A \Rightarrow B \equiv \neg A \lor B \]

Applying this to our expression:
\[ \neg (\neg q \lor r) \lor p \]

### Step 5: Apply De Morgan's Law
Using De Morgan's Law, we can simplify \( \neg (\neg q \lor r) \):
\[ \neg (\neg q \lor r) \equiv q \land \neg r \]

So, we have:
\[ (q \land \neg r) \lor p \]

### Final Answer
The converse of the statement \( p \Rightarrow (q \Rightarrow r) \) is:
\[ (q \land \neg r) \lor p \]