If p and q are logical statements, then `p rArr (~q rArr p)` is equivalent to

Options

A. `prArr(prArrq)`
B. `p rArr (p^^q)`
C. `prArr (p^^q)`
D. `p rArr (phArr q)`

Answer

To solve the logical statement \( p \Rightarrow (\neg q \Rightarrow p) \) and find its equivalent form, we can follow these steps:

### Step 1: Rewrite the implication
We know that an implication \( a \Rightarrow b \) can be rewritten as \( \neg a \lor b \). Therefore, we can rewrite \( \neg q \Rightarrow p \) as:
\[ \neg (\neg q) \lor p \]
This simplifies to:
\[ q \lor p \]

### Step 2: Substitute back into the original statement
Now, we substitute this back into the original expression:
\[ p \Rightarrow (q \lor p) \]

### Step 3: Rewrite the outer implication
Again, we apply the implication transformation:
\[ p \Rightarrow (q \lor p) \equiv \neg p \lor (q \lor p) \]

### Step 4: Simplify the expression
Using the associative property of logical disjunction, we can rearrange the terms:
\[ \neg p \lor q \lor p \]
This can be further simplified to:
\[ (\neg p \lor p) \lor q \]
Since \( \neg p \lor p \) is a tautology (always true), we have:
\[ \text{True} \lor q \equiv \text{True} \]

### Step 5: Determine the equivalent expression
The expression \( p \Rightarrow (\neg q \Rightarrow p) \) simplifies to \( \neg p \lor (q \lor p) \), which is logically equivalent to \( p \lor q \).

### Conclusion
Thus, the equivalent expression for \( p \Rightarrow (\neg q \Rightarrow p) \) is:
\[ p \lor q \]

### Final Answer
The correct option is **B: \( p \Rightarrow (p \lor q) \)**.