The statement `p rarr (q rarr p)` is equivalent to

Options

A. `p rarr (p vv q)`
B. `p rarr (p ^^ q)`
C. `p rarr (p ltrarr q)`
D. `p rarr (p rarr q)`

Answer

To determine the equivalence of the statement \( p \rightarrow (q \rightarrow p) \), we will follow a systematic approach using logical equivalences.

### Step-by-Step Solution:

1. **Understand the Implication**: The statement \( p \rightarrow (q \rightarrow p) \) can be rewritten using the definition of implication. Recall that \( A \rightarrow B \) is equivalent to \( \neg A \lor B \). Thus, we can rewrite the statement as:
\[ p \rightarrow (q \rightarrow p) \equiv \neg p \lor (q \rightarrow p) \]

2. **Rewrite the Inner Implication**: Now, we need to rewrite \( q \rightarrow p \) using the same definition:
\[ q \rightarrow p \equiv \neg q \lor p \]
Substituting this back into our expression gives:
\[ \neg p \lor (q \rightarrow p) \equiv \neg p \lor (\neg q \lor p) \]

3. **Associative Law of Disjunction**: We can apply the associative law of disjunction, which allows us to regroup the terms:
\[ \neg p \lor (\neg q \lor p) \equiv (\neg p \lor p) \lor \neg q \]

4. **Apply the Law of Excluded Middle**: The expression \( \neg p \lor p \) is always true (this is known as the law of excluded middle):
\[ \neg p \lor p \equiv \text{True} \]
Therefore, we can simplify our expression to:
\[ \text{True} \lor \neg q \]

5. **Final Simplification**: Since the disjunction of any statement with true is always true, we have:
\[ \text{True} \lor \neg q \equiv \text{True} \]

Thus, we conclude that:
\[ p \rightarrow (q \rightarrow p) \equiv \text{True} \]

### Conclusion:
The statement \( p \rightarrow (q \rightarrow p) \) is equivalent to True.