Key Concepts and Formulas

• Implication: $P \Rightarrow Q$ is equivalent to $\sim P \vee Q$.
• Double Negation: $\sim (\sim P)$ is equivalent to $P$.
• Distributive Law: $P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)$ and $P \wedge (Q \vee R) \equiv (P \wedge Q) \vee (P \wedge R)$.
• Associative Law: $(P \vee Q) \vee R \equiv P \vee (Q \vee R)$ and $(P \wedge Q) \wedge R \equiv P \wedge (Q \wedge R)$.
• Commutative Law: $P \vee Q \equiv Q \vee P$ and $P \wedge Q \equiv Q \wedge P$.

Step-by-Step Solution

Step 1: Convert the given statement into its equivalent form using the implication rule.

The given statement is $B \Rightarrow ((\sim A) \vee B)$. Using the implication rule, $P \Rightarrow Q \equiv \sim P \vee Q$, we have: $B \Rightarrow ((\sim A) \vee B) \equiv \sim B \vee ((\sim A) \vee B)$

Step 2: Simplify the expression using the associative law.

Since the expression now only contains the 'or' operator, we can use the associative law: $\sim B \vee ((\sim A) \vee B) \equiv (\sim B \vee B) \vee (\sim A)$

Step 3: Simplify using the fact that $P \vee (\sim P)$ is always true.

Since $B \vee (\sim B)$ is a tautology (always true), we replace it with $T$: $(\sim B \vee B) \vee (\sim A) \equiv T \vee (\sim A)$

Step 4: Simplify using the fact that $T \vee P$ is always true.

Since $T \vee (\sim A)$ is always true, the entire statement is a tautology: $T \vee (\sim A) \equiv T$

Step 5: Analyze option (A) and convert it to its equivalent form using the implication rule.

Option (A) is $B \Rightarrow ((\sim A) \Rightarrow B)$. Using the implication rule on $(\sim A) \Rightarrow B$, we get $\sim (\sim A) \vee B \equiv A \vee B$. So, the expression becomes $B \Rightarrow (A \vee B)$. Using the implication rule again, we get $\sim B \vee (A \vee B)$.

Step 6: Simplify the expression from option (A) using the associative law.

Since the expression now only contains the 'or' operator, we can use the associative law: $\sim B \vee (A \vee B) \equiv (\sim B \vee B) \vee A$

Step 7: Simplify using the fact that $P \vee (\sim P)$ is always true.

Since $B \vee (\sim B)$ is a tautology (always true), we replace it with $T$: $(\sim B \vee B) \vee A \equiv T \vee A$

Step 8: Simplify using the fact that $T \vee P$ is always true.

Since $T \vee A$ is always true, the entire statement is a tautology: $T \vee A \equiv T$

Step 9: Compare the simplified forms.

The original expression simplified to $T$, and option (A) also simplified to $T$. Therefore, the original statement is equivalent to option (A).

Common Mistakes & Tips

• Remember the implication rule correctly: $P \Rightarrow Q \equiv \sim P \vee Q$. A common mistake is to forget the negation.
• When simplifying, be systematic. Apply one rule at a time.
• Recognize tautologies and contradictions to simplify expressions efficiently.

Summary

We started by simplifying the given statement $B \Rightarrow ((\sim A) \vee B)$ using the implication rule and associative law, eventually arriving at a tautology, $T$. We then simplified option (A), $B \Rightarrow ((\sim A) \Rightarrow B)$, using the same rules, also arriving at $T$. Thus, the given statement is equivalent to option (A).

Final Answer

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