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, we will use a truth table approach.

### Step-by-Step Solution:

1. **Identify the Variables:**
We have two logical statements: \( p \) and \( q \).

2. **Construct the Truth Table:**
We will create a truth table for \( p \), \( q \), \( \neg q \), and \( \neg q \Rightarrow p \).

| \( p \) | \( q \) | \( \neg q \) | \( \neg q \Rightarrow p \) | \( p \Rightarrow (\neg q \Rightarrow p) \) |
|---------|---------|---------------|-----------------------------|-------------------------------------------|
| T | T | F | T | T |
| T | F | T | T | T |
| F | T | F | T | T |
| F | F | T | F | T |

3. **Evaluate \( \neg q \):**
- If \( q \) is true (T), then \( \neg q \) is false (F).
- If \( q \) is false (F), then \( \neg q \) is true (T).

4. **Evaluate \( \neg q \Rightarrow p \):**
- The implication \( A \Rightarrow B \) is false only when \( A \) is true and \( B \) is false.
- From the table, we see that \( \neg q \Rightarrow p \) is true in all cases except when \( p \) is false and \( \neg q \) is true.

5. **Evaluate \( p \Rightarrow (\neg q \Rightarrow p) \):**
- This statement is true in all cases, as shown in the last column of the truth table.

6. **Determine the Equivalent Statement:**
Since the truth table for \( p \Rightarrow (\neg q \Rightarrow p) \) is always true, we can conclude that this statement is a tautology.

7. **Check the Options:**
We need to check which of the provided options matches the truth table we derived.

- **Option 1:** \( p \Rightarrow (p \Rightarrow q) \) - Not equivalent.
- **Option 2:** \( p \land (p \Rightarrow q) \) - Not equivalent.
- **Option 3:** \( p \Rightarrow (p \lor q) \) - Equivalent.
- **Option 4:** \( p \Rightarrow (p \iff q) \) - Not equivalent.

### Conclusion:
The equivalent statement for \( p \Rightarrow (\neg q \Rightarrow p) \) is **Option 3: \( p \Rightarrow (p \lor q) \)**.