The statement $$p \rightarrow (q \rightarrow p)$$ is equivalent to :

We are given the statement $$ p \rightarrow (q \rightarrow p) $$ and need to find which option it is equivalent to. Recall that the implication $$ a \rightarrow b $$ is logically equivalent to $$ \neg a \vee b $$. We will use this equivalence to simplify the given statement step by step.

First, consider the inner implication $$ q \rightarrow p $$. Using the equivalence, we rewrite it as $$ \neg q \vee p $$. So the entire statement becomes:

$$ p \rightarrow (\neg q \vee p) $$

Now, apply the equivalence to the outer implication $$ p \rightarrow (\neg q \vee p) $$. This becomes:

$$ \neg p \vee (\neg q \vee p) $$

Disjunction (OR) is associative, meaning we can regroup the terms without changing the meaning. So we write:

$$ (\neg p \vee p) \vee \neg q $$

Notice that $$ \neg p \vee p $$ is a tautology, meaning it is always true, regardless of the truth value of $$ p $$. Therefore:

$$ \neg p \vee p = \text{true} $$

Substituting this, we get:

$$ \text{true} \vee \neg q $$

The disjunction of true and any statement is always true. So:

$$ \text{true} \vee \neg q = \text{true} $$

Thus, the original statement $$ p \rightarrow (q \rightarrow p) $$ simplifies to true, meaning it is a tautology (always true).

Now, we check each option to see which one is also a tautology and equivalent to true.

Option A: $$ p \rightarrow q $$

Rewrite using equivalence:

$$ \neg p \vee q $$

This is not always true. For example, when $$ p $$ is true and $$ q $$ is false, $$ \neg p \vee q = \text{false} \vee \text{false} = \text{false} $$. So it is not a tautology and not equivalent to the original statement.

Option B: $$ p \rightarrow (p \vee q) $$

Rewrite using equivalence:

$$ \neg p \vee (p \vee q) $$

Associativity allows regrouping:

$$ (\neg p \vee p) \vee q $$

Again, $$ \neg p \vee p = \text{true} $$, so:

$$ \text{true} \vee q = \text{true} $$

This is a tautology. Therefore, it is equivalent to the original statement.

Option C: $$ p \rightarrow (p \rightarrow q) $$

First, rewrite the inner implication $$ p \rightarrow q $$ as $$ \neg p \vee q $$. So the statement becomes:

$$ p \rightarrow (\neg p \vee q) $$

Apply equivalence to the outer implication:

$$ \neg p \vee (\neg p \vee q) $$

Associativity and idempotence (since $$ \neg p \vee \neg p = \neg p $$):

$$ (\neg p \vee \neg p) \vee q = \neg p \vee q $$

This is $$ \neg p \vee q $$, which is the same as $$ p \rightarrow q $$. As in Option A, this is not a tautology (e.g., false when $$ p $$ is true and $$ q $$ is false). So it is not equivalent to the original statement.

Option D: $$ p \rightarrow (p \wedge q) $$

Rewrite using equivalence:

$$ \neg p \vee (p \wedge q) $$

Distribute $$ \vee $$ over $$ \wedge $$:

$$ (\neg p \vee p) \wedge (\neg p \vee q) $$

Now, $$ \neg p \vee p = \text{true} $$, so:

$$ \text{true} \wedge (\neg p \vee q) = \neg p \vee q $$

Again, this is $$ \neg p \vee q $$, same as $$ p \rightarrow q $$, which is not a tautology. So it is not equivalent to the original statement.

Only Option B simplifies to true, making it a tautology and equivalent to the original statement $$ p \rightarrow (q \rightarrow p) $$.

Hence, the correct answer is Option B.