Key Concepts and Formulas

• Logical OR (∨): The statement $A \vee B$ is true if at least one of $A$ or $B$ is true. It is only false if both $A$ and $B$ are false.
• Logical NOT (∼): The statement $\sim A$ is true if $A$ is false, and vice versa.
• Conditional Statement (→): The statement $A \to B$ is equivalent to $(\sim A) \vee B$. The statement $A \to B$ is only false when $A$ is true and $B$ is false.
• Associative Law for OR: $(A \vee B) \vee C \equiv A \vee (B \vee C)$. This means we can drop parentheses when dealing with only OR operations.

Step-by-Step Solution

Step 1: Simplify S₁

Our goal is to simplify $S_1$ using the associative property of the OR operator. $S_1: ((\sim p) \vee q) \vee ((\sim p) \vee r)$ Since the OR operation is associative, we can rewrite $S_1$ as: $S_1: (\sim p) \vee q \vee (\sim p) \vee r$ Since OR operation is idempotent ($A \vee A \equiv A$), we can further simplify by combining the $\sim p$ terms: $S_1: (\sim p) \vee (\sim p) \vee q \vee r$ $S_1: (\sim p) \vee q \vee r$

Step 2: Simplify S₂

Our goal is to rewrite $S_2$ using the definition of the conditional statement. $S_2: p \to (q \vee r)$ Using the equivalence $A \to B \equiv (\sim A) \vee B$, we can rewrite $S_2$ as: $S_2: (\sim p) \vee (q \vee r)$ Due to the associative property of OR, we can rewrite it as: $S_2: (\sim p) \vee q \vee r$

Step 3: Compare S₁ and S₂

We have simplified both $S_1$ and $S_2$: $S_1: (\sim p) \vee q \vee r$ $S_2: (\sim p) \vee q \vee r$ Therefore, $S_1$ and $S_2$ are logically equivalent. This means they always have the same truth value. If $S_1$ is true, $S_2$ is true, and if $S_1$ is false, $S_2$ is false.

Step 4: Analyze the Options

Now we analyze each option to determine which one is NOT true:

• (A) If S₂ is True, then S₁ is True: Since $S_1$ and $S_2$ are equivalent, this statement is true.
• (B) If S₂ is False, then S₁ is False: Since $S_1$ and $S_2$ are equivalent, this statement is true.
• (C) If S₂ is False, then S₁ is True: Since $S_1$ and $S_2$ are equivalent, if $S_2$ is false, then $S_1$ must also be false. Therefore, this statement is false.
• (D) If S₁ is False, then S₂ is False: Since $S_1$ and $S_2$ are equivalent, this statement is true.

Step 5: Identify the Incorrect Option

We found that option (C) is NOT true.

Common Mistakes & Tips

• Confusion with Logical Equivalences: Remember the fundamental equivalences, particularly $A \to B \equiv (\sim A) \vee B$.
• Associativity and Commutativity: Recognize and utilize the associative and commutative properties of logical operators to simplify expressions.
• Truth Tables: When in doubt, constructing a truth table can help verify the equivalence of logical statements.

Summary

We simplified the given compound statements $S_1$ and $S_2$ using logical equivalences and the associative property of the OR operator. We found that $S_1$ and $S_2$ are logically equivalent. Therefore, statements (B) and (D) are true. Statement (A) is also true. Statement (C) is false, which is the answer.

Final Answer

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