The given statement is \( p \rightarrow (q \rightarrow p) \). We need to find its equivalent logical statement among the given options. Let us analyze each option step by step.
The statement \( q \rightarrow p \) is a logical implication that can be expressed in terms of logical operators as \( \neg q \vee p \). So, the original statement \( p \rightarrow (q \rightarrow p) \) can be rewritten as:
\( p \rightarrow (\neg q \vee p) \).
According to the definition of logical implication, \( p \rightarrow (\neg q \vee p) \) is equivalent to:
- Replace the implication with disjunction and negation: \( \neg p \vee (\neg q \vee p) \).
- Apply associative law: \( (\neg p \vee \neg q) \vee p \).
- Apply the law of disjunction: \( p \vee \neg q \), which simplifies further to \( p \vee q \) given the nature of implication and disjunction laws.
This simplified form shows that the statement \( p \rightarrow (q \rightarrow p) \) is equivalent to \( p \rightarrow (p \vee q) \).
Let's verify this result by evaluating the options:
- Option 1: \( p \rightarrow (p \leftrightarrow q) \) is not equivalent since \( p \leftrightarrow q \) implies equality which diverges from the derived equivalence.
- Option 2: \( p \rightarrow (p \rightarrow q) \) transforms to \( \neg p \vee (\neg p \vee q) = \neg p \vee q \), which again is not equivalent.
- Option 3: \( p \rightarrow (p \vee q) \) which is generally in line with our result. So this option is equivalent.
- Option 4: \( p \rightarrow (p \wedge q) = \neg p \vee (p \wedge q)\) is not equivalent as conjunction follows stricter rules of logical dependence.
The correct equivalent statement is the option \( p \rightarrow (p \vee q) \).