Key Concepts and Formulas

• Biconditional (Equivalence): $p \leftrightarrow q$ is true when both $p$ and $q$ have the same truth value (both true or both false). It's false when they have different truth values.
• Negation: $\sim p$ is true when $p$ is false, and vice versa.
• Tautology: A statement that is always true, regardless of the truth values of its components.
• De Morgan's Law for Equivalence: $\sim(p \leftrightarrow q) \equiv (p \leftrightarrow \sim q) \equiv (\sim p \leftrightarrow q)$

Step-by-Step Solution

Step 1: Analyze Statement-1

We want to determine if $\sim(p \leftrightarrow \sim q)$ is equivalent to $p \leftrightarrow q$. Let's construct a truth table for both expressions.

Step 2: Construct the Truth Table for $\sim(p \leftrightarrow \sim q)$

Step 3: Construct the Truth Table for $p \leftrightarrow q$

Step 4: Compare the Truth Tables

The truth table for $\sim(p \leftrightarrow \sim q)$ is identical to the truth table for $p \leftrightarrow q$. Therefore, $\sim(p \leftrightarrow \sim q) \equiv (p \leftrightarrow q)$. Statement-1 is true.

Step 5: Analyze Statement-2

Statement-2 claims that $\sim(p \leftrightarrow \sim q)$ is a tautology. A tautology is always true. From the truth table in Step 2, we see that $\sim(p \leftrightarrow \sim q)$ is not always true (it is false in some cases). Therefore, Statement-2 is false.

Step 6: Re-evaluate Statement 2

Statement 2 incorrectly claims $\sim(p \leftrightarrow \sim q)$ is a tautology. Since we established in Step 4 that $\sim(p \leftrightarrow \sim q)$ is equivalent to $p \leftrightarrow q$, and $p \leftrightarrow q$ is not a tautology, Statement 2 is false.

Step 7: Re-evaluate Statement 1

From the truth tables, $\sim(p \leftrightarrow \sim q)$ is equivalent to $p \leftrightarrow q$. Thus, Statement 1 is true.

Step 8: Check if Statement 2 explains Statement 1

Since Statement 2 is false, it cannot be a correct explanation for Statement 1.

Step 9: Determine the correct option

Statement-1 is true, and Statement-2 is false.

Common Mistakes & Tips

• Carefully construct the truth tables. A single error can lead to the wrong conclusion.
• Remember the definition of a tautology.
• Understand the meaning of equivalence ($\leftrightarrow$) and negation ($\sim$).
• Be cautious with negations, especially when dealing with compound statements.

Summary

We analyzed the given statements by constructing truth tables. We found that $\sim(p \leftrightarrow \sim q)$ is equivalent to $p \leftrightarrow q$, making Statement-1 true. Statement-2, claiming that $\sim(p \leftrightarrow \sim q)$ is a tautology, is false because its truth table has both true and false entries. Since Statement 2 is false, it cannot explain Statement 1. Therefore, Statement-1 is true, Statement-2 is false. Option (C) is incorrect. The correct option is (A) because Statement 1 is true, Statement 2 is true, and Statement 2 is a correct explanation for Statement 1. However, this is incorrect. It must be option (C) instead.

Final Answer

The final answer is \boxed{C}.