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 solve the problem, we need to find the logical equivalence of the expression \( P \rightarrow (Q \rightarrow P) \). We'll do this by constructing truth tables for the given expression and comparing it with other logical expressions.

### Step 1: Construct the truth table for \( P \) and \( Q \)

We start by listing all possible truth values for the logical statements \( P \) and \( Q \):

| \( P \) | \( Q \) |
|---------|---------|
| T | T |
| T | F |
| F | T |
| F | F |

### Step 2: Evaluate \( Q \rightarrow P \)

Now we evaluate the expression \( Q \rightarrow P \). Recall that \( Q \rightarrow P \) is true except when \( Q \) is true and \( P \) is false.

| \( P \) | \( Q \) | \( Q \rightarrow P \) |
|---------|---------|-----------------------|
| T | T | T |
| T | F | T |
| F | T | F |
| F | F | T |

### Step 3: Evaluate \( P \rightarrow (Q \rightarrow P) \)

Next, we evaluate the entire expression \( P \rightarrow (Q \rightarrow P) \). We will use the results from the previous step.

| \( P \) | \( Q \) | \( Q \rightarrow P \) | \( P \rightarrow (Q \rightarrow P) \) |
|---------|---------|-----------------------|---------------------------------------|
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |

### Step 4: Analyze the results

From the truth table, we see that \( P \rightarrow (Q \rightarrow P) \) is true in all cases, which means it is a tautology.

### Step 5: Compare with other logical expressions

Now we need to compare this result with other logical statements to find an equivalent expression.

1. **Option A**: \( P \rightarrow Q \)
2. **Option B**: \( P \rightarrow (P \lor Q) \)
3. **Option C**: \( P \rightarrow (P \land Q) \)
4. **Option D**: \( P \rightarrow (P \equiv Q) \)

We will construct truth tables for each of these options and compare them with the truth table of \( P \rightarrow (Q \rightarrow P) \).

### Step 6: Evaluate \( P \rightarrow (P \lor Q) \)

| \( P \) | \( Q \) | \( P \lor Q \) | \( P \rightarrow (P \lor Q) \) |
|---------|---------|----------------|---------------------------------|
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | T |

### Conclusion

Since the truth table for \( P \rightarrow (P \lor Q) \) matches the truth table for \( P \rightarrow (Q \rightarrow P) \), we can conclude that:

\[ P \rightarrow (Q \rightarrow P) \text{ is equivalent to } P \rightarrow (P \lor Q) \]

Thus, the correct answer is **Option B**.