Key Concepts and Formulas

• Implication: $A \to B \equiv \sim A \vee B$ (A implies B is equivalent to not A or B)
• Commutative Law: $A \vee B \equiv B \vee A$ and $A \wedge B \equiv B \wedge A$
• Associative Law: $(A \vee B) \vee C \equiv A \vee (B \vee C)$ and $(A \wedge B) \wedge C \equiv A \wedge (B \wedge C)$
• Tautology: $A \vee \sim A \equiv t$ (always true)
• Identity Law: $A \vee t \equiv t$ and $A \wedge f \equiv f$
• Equivalence (Biconditional): $A \leftrightarrow B \equiv (A \to B) \wedge (B \to A)$

Step-by-Step Solution

Step 1: Simplify the given statement using the implication rule.

We are given the statement $A \to (B \to A)$. We use the implication rule $X \to Y \equiv \sim X \vee Y$ to rewrite the inner implication.

$A \to (B \to A) \equiv A \to (\sim B \vee A)$

Step 2: Apply the implication rule again.

We apply the implication rule again to remove the outer implication.

$A \to (\sim B \vee A) \equiv \sim A \vee (\sim B \vee A)$

Step 3: Rearrange the terms using the commutative and associative laws.

We use the commutative and associative laws to rearrange the terms to group $A$ and $\sim A$ together.

$\sim A \vee (\sim B \vee A) \equiv \sim A \vee (A \vee \sim B) \equiv (\sim A \vee A) \vee \sim B$

Step 4: Simplify using the tautology rule.

We simplify the expression using the tautology rule $A \vee \sim A \equiv t$.

$(\sim A \vee A) \vee \sim B \equiv t \vee \sim B$

Step 5: Simplify using the identity law.

We simplify the expression using the identity law $t \vee A \equiv t$.

$t \vee \sim B \equiv t$

So, the statement $A \to (B \to A)$ is equivalent to a tautology.

Step 6: Check the options.

We need to find an option that is also equivalent to a tautology. Let's analyze option (B): $A \to (A \vee B)$.

$A \to (A \vee B) \equiv \sim A \vee (A \vee B)$

Step 7: Rearrange the terms using the associative law.

We use the associative law to group $A$ and $\sim A$ together.

$\sim A \vee (A \vee B) \equiv (\sim A \vee A) \vee B$

Step 8: Simplify using the tautology rule.

We simplify the expression using the tautology rule $A \vee \sim A \equiv t$.

$(\sim A \vee A) \vee B \equiv t \vee B$

Step 9: Simplify using the identity law.

We simplify the expression using the identity law $t \vee A \equiv t$.

$t \vee B \equiv t$

So, the statement $A \to (A \vee B)$ is also equivalent to a tautology.

Common Mistakes & Tips

• Remember the implication rule correctly: $A \to B \equiv \sim A \vee B$. A common mistake is to write it as $\sim A \wedge B$ or $A \vee \sim B$.
• When simplifying, use the commutative and associative laws to rearrange terms to group complementary statements together (e.g., $A$ and $\sim A$).
• Recognize tautologies and contradictions quickly to simplify expressions.

Summary

We simplified the given statement $A \to (B \to A)$ to a tautology $t$. Then we checked the given options to find an equivalent statement. Option (B), $A \to (A \vee B)$, also simplifies to a tautology $t$. Therefore, the correct answer is option (B).

Final Answer

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