The statement `~prarr(qrarrp)` is equivalent to

Options

A. `prarr(prarrq)`
B. `prarr(p vv q)`
C. `q rarrp`
D. `q rarr (prarrq)`

Answer

To determine the equivalence of the statement `~p → (q → p)`, we will use a truth table to analyze the logical values of the components involved.

### Step-by-Step Solution:

1. **Identify Variables**:
- Let \( p \) and \( q \) be the two propositions.

2. **Construct the Truth Table**:
- We will create a truth table that includes the columns for \( p \), \( q \), \( \neg p \) (negation of \( p \)), \( p \rightarrow q \) (implication from \( p \) to \( q \)), and \( q \rightarrow p \) (implication from \( q \) to \( p \)).

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

3. **Calculate \( \neg p \)**:
- Negation of \( p \) is true when \( p \) is false and vice versa.

4. **Calculate \( p \rightarrow q \)**:
- This implication is false only when \( p \) is true and \( q \) is false.

5. **Calculate \( q \rightarrow p \)**:
- This implication is false only when \( q \) is true and \( p \) is false.

6. **Evaluate the Statement**:
- We need to evaluate \( \neg p \rightarrow (q \rightarrow p) \).
- This will be true unless \( \neg p \) is true and \( q \rightarrow p \) is false.

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

7. **Final Evaluation**:
- The final column shows the truth values for \( \neg p \rightarrow (q \rightarrow p) \). The statement is false only when \( p \) is false and \( q \) is true.

8. **Compare with Options**:
- Now we need to check which of the options matches this truth table.
- After checking all options, we find that option C, which is \( q \rightarrow p \), matches the truth values of our original statement.

### Conclusion:
The statement `~p → (q → p)` is equivalent to option C, which is \( q \rightarrow p \).