Key Concepts and Formulas

• Conditional Statement: $p \to q$ is equivalent to $\neg p \vee q$ (If $p$ then $q$ is true if $p$ is false or $q$ is true).
• Equivalence: Two statements are equivalent if they have the same truth table.
• Truth Table: A table showing all possible truth values of a statement.

Step-by-Step Solution

Step 1: Simplify $p \to (q \to p)$ using the conditional statement equivalence. We are given the statement $p \to (q \to p)$. Using the equivalence $a \to b \equiv \neg a \vee b$, we can rewrite the inner conditional as: $p \to (q \to p) \equiv p \to (\neg q \vee p)$

Step 2: Simplify the expression further using the conditional statement equivalence again. Applying the conditional equivalence to the entire expression, we get: $p \to (\neg q \vee p) \equiv \neg p \vee (\neg q \vee p)$

Step 3: Rearrange the terms using the associative property of the "or" operation. The "or" operation is associative, meaning $(a \vee b) \vee c \equiv a \vee (b \vee c)$. Thus, we can rearrange the terms as: $\neg p \vee (\neg q \vee p) \equiv (\neg p \vee p) \vee \neg q$

Step 4: Simplify $(\neg p \vee p)$ using the law of excluded middle. The law of excluded middle states that $p \vee \neg p$ is always true (T). Therefore: $(\neg p \vee p) \vee \neg q \equiv T \vee \neg q$

Step 5: Simplify $T \vee \neg q$ using the properties of "or" with True. Since anything "or" True is True, we have: $T \vee \neg q \equiv T$

Step 6: Analyze the options to find an equivalent statement to $T$. Now we need to check which of the options simplifies to T. Let's analyze option (C) $p \to (p \vee q)$. $p \to (p \vee q) \equiv \neg p \vee (p \vee q) \equiv (\neg p \vee p) \vee q \equiv T \vee q \equiv T$ Thus, option (C) is equivalent to the original statement. However, the provided answer is (A), so we must have made an error. Let's re-examine the options.

Step 7: Analyze the options to find an equivalent statement to $p \to (p \leftrightarrow q)$. Let's examine option (A) $p \to (p \leftrightarrow q)$. First, we know that $p \leftrightarrow q \equiv (p \to q) \wedge (q \to p)$. Therefore, $p \to (p \leftrightarrow q) \equiv p \to ((p \to q) \wedge (q \to p)) \equiv p \to ((\neg p \vee q) \wedge (\neg q \vee p))$ This looks complicated. Since we know the answer is T, let's see if we can prove that directly.

Step 8: Reconsider the original problem and the given correct answer. The original statement is $p \to (q \to p)$. The answer given is $p \to (p \leftrightarrow q)$. We want to show that $p \to (q \to p) \equiv p \to (p \leftrightarrow q)$. This means we want to show that $\neg p \vee (\neg q \vee p) \equiv \neg p \vee ((p \to q) \wedge (q \to p))$ Since $\neg p \vee (\neg q \vee p) = (\neg p \vee p) \vee \neg q = T \vee \neg q = T$, we want to show that $\neg p \vee (p \leftrightarrow q)$ is always true. Let's analyze the truth table for $p \to (p \leftrightarrow q)$:

This is NOT always true. There must be an error in the stated correct answer. Based on our simplification, $p \to (q \to p)$ is equivalent to True. Therefore ANY expression that is always true is equivalent. Option (C), $p \to (p \vee q)$ is always true, as shown in Step 6.

Step 9: The correct answer must be (C). There is an error in the provided correct answer. The correct answer should be (C), $p \to (p \vee q)$. Since the problem statement requires us to arrive at the stated correct answer, let's try to manipulate the expression to look like the given option (A).

Step 10: Working backwards from option (A) $p \to (p \leftrightarrow q)$. $p \to (p \leftrightarrow q) \equiv \neg p \vee (p \leftrightarrow q) \equiv \neg p \vee ((p \to q) \wedge (q \to p)) \equiv \neg p \vee ((\neg p \vee q) \wedge (\neg q \vee p))$ We know that $p \to (q \to p) \equiv \neg p \vee (\neg q \vee p) = T$. We want to show that $\neg p \vee ((\neg p \vee q) \wedge (\neg q \vee p)) = T$. Since $T = \neg p \vee (\neg q \vee p)$, we want to show that $\neg p \vee ((\neg p \vee q) \wedge (\neg q \vee p)) = \neg p \vee (\neg q \vee p)$. This is NOT true in general.

Step 11: Examining the truth table of $p \to (q \to p)$

Step 12: Re-evaluating the options.

Since $p \to (q \to p)$ is always true, any option that is always true will be equivalent. Let's check each option. (A) $p \to (p \leftrightarrow q)$ - Not always true. (B) $p \to (p \to q)$ - $p \to (\neg p \vee q) \equiv \neg p \vee (\neg p \vee q) \equiv \neg p \vee q$. Not always true. (C) $p \to (p \vee q)$ - $\neg p \vee (p \vee q) \equiv (\neg p \vee p) \vee q \equiv T \vee q \equiv T$. Always true. (D) $p \to (p \wedge q)$ - $\neg p \vee (p \wedge q) \equiv (\neg p \vee p) \wedge (\neg p \vee q) \equiv T \wedge (\neg p \vee q) \equiv \neg p \vee q$. Not always true.

The only expression that is always true is (C). There is an error in the problem statement.

Step 13: Let's assume the answer is (A) and try to prove it. Assume $p \to (p \leftrightarrow q)$ is the correct answer. Then $p \to (q \to p) \equiv p \to (p \leftrightarrow q)$. This means $\neg p \vee (\neg q \vee p) \equiv \neg p \vee ((p \to q) \wedge (q \to p))$. Simplifying, $T \equiv \neg p \vee ((\neg p \vee q) \wedge (\neg q \vee p))$. Since $T$ is always true, $\neg p \vee ((\neg p \vee q) \wedge (\neg q \vee p))$ must also be always true. This is NOT true.

Common Mistakes & Tips

• Be careful with the order of operations and the equivalences of logical statements.
• When in doubt, construct a truth table to verify the equivalence of statements.
• Double-check your application of De Morgan's laws and other logical identities.

Summary

We simplified the given expression $p \to (q \to p)$ using logical equivalences to $T$ (True). Among the options, only option (C), $p \to (p \vee q)$, also simplifies to $T$. Therefore, the correct answer is (C). The problem statement incorrectly states that the correct answer is (A).

Final Answer

The final answer is \boxed{C}, which corresponds to option (C).