Key Concepts and Formulas

• Implication: $p \to q \equiv \sim p \vee q$
• Disjunction: $p \vee q$ is true if at least one of $p$ or $q$ is true.
• Tautology: A statement that is always true.

Step-by-Step Solution

Step 1: Rewrite the given expression using the implication rule.

We are given the statement $(p \to (q \to p)) \to (p \to (p \vee q))$. We want to simplify this expression using logical equivalences to determine if it's a tautology, contradiction, or equivalent to another expression. We start by replacing the implications with their equivalent disjunction forms.

$(p \to (q \to p)) \to (p \to (p \vee q)) \equiv (\sim p \vee (q \to p)) \to (\sim p \vee (p \vee q))$

Step 2: Apply the implication rule again.

We apply the implication rule to the term $(q \to p)$ inside the first parenthesis.

$(\sim p \vee (q \to p)) \to (\sim p \vee (p \vee q)) \equiv (\sim p \vee (\sim q \vee p)) \to (\sim p \vee (p \vee q))$

Step 3: Simplify the expressions within the parentheses using the commutative and associative properties of disjunction.

We can rearrange the terms within the first set of parentheses. Also, the associative property allows us to remove the inner parentheses.

$(\sim p \vee \sim q \vee p) \to (\sim p \vee p \vee q)$

Step 4: Simplify using the fact that $p \vee \sim p$ is always true.

We know that $p \vee \sim p$ is always true, so we can replace $(\sim p \vee p)$ with $T$ (True).

$(\sim p \vee p \vee \sim q) \to (\sim p \vee p \vee q) \equiv (T \vee \sim q) \to (T \vee q)$

Step 5: Simplify using the fact that $T \vee x$ is always true.

Since $T \vee x$ is always true for any $x$, both sides of the implication become $T$.

$(T \vee \sim q) \to (T \vee q) \equiv T \to T$

Step 6: Evaluate the implication.

An implication $T \to T$ is always true.

$T \to T \equiv T$

Therefore, the original statement is a tautology.

Common Mistakes & Tips

• Be careful when applying the implication rule. Remember that $p \to q$ is equivalent to $\sim p \vee q$, not $p \vee \sim q$.
• When simplifying, use the commutative and associative properties to group terms that can be easily reduced (like $p \vee \sim p$).
• Recognize common tautologies like $p \vee \sim p$ and $p \to p$ to simplify the expression quickly.

Summary

We started with the given statement and repeatedly applied the implication rule and simplified using logical equivalences. By rewriting the implications as disjunctions and utilizing the properties of disjunction and the tautology $p \vee \sim p$, we were able to reduce the expression to $T \to T$, which is always true. Therefore, the original statement is a tautology.

Final Answer

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