If p and q are two logical statements, then `prArr (qrArrp)` is equivalent to

Options

A. `prArr (p rArr q)`
B. `p rArr (p vv q)`
C. `p rArr (p ^^ q)`
D. `prArr (p hArr q)`

Answer

To determine the equivalence of the logical statement \( p \rightarrow (q \rightarrow p) \), we can use a truth table. Let's go through the steps systematically.

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

### Step 2: Create the Truth Table
Since we have two variables, the number of possible combinations of truth values is \( 2^2 = 4 \). We will list all combinations of truth values for \( p \) and \( q \).

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

### Step 3: Calculate \( q \rightarrow p \)
The implication \( q \rightarrow p \) is false only when \( q \) is true and \( p \) is false. We will calculate this for each row.

| \( p \) | \( q \) | \( q \rightarrow p \) |
|---------|---------|------------------------|
| T | T | T |
| T | F | T |
| F | T | F |
| F | F | T |

### Step 4: Calculate \( p \rightarrow (q \rightarrow p) \)
Now we will calculate \( p \rightarrow (q \rightarrow p) \). This implication is false only when \( p \) is true and \( (q \rightarrow p) \) is false.

| \( p \) | \( q \) | \( q \rightarrow p \) | \( p \rightarrow (q \rightarrow p) \) |
|---------|---------|------------------------|----------------------------------------|
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |

### Step 5: Final Truth Table
The final truth table for \( p \rightarrow (q \rightarrow p) \) is:

| \( p \) | \( q \) | \( p \rightarrow (q \rightarrow p) \) |
|---------|---------|----------------------------------------|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | T |

### Conclusion
The expression \( p \rightarrow (q \rightarrow p) \) is always true regardless of the truth values of \( p \) and \( q \). Thus, it is equivalent to the logical statement "True".