Key Concepts and Formulas

• Logical Connectives: Understanding the truth tables for AND ($\wedge$), OR ($\vee$), NOT ($\sim$), implication ($\to$), and bi-conditional ($\leftrightarrow$).
• Tautology: A statement that is always true, regardless of the truth values of its components.
• Fallacy (Contradiction): A statement that is always false, regardless of the truth values of its components.
• Implication: $p \to q$ is equivalent to $\sim p \vee q$.
• Bi-conditional: $p \leftrightarrow q$ is equivalent to $(p \to q) \wedge (q \to p)$.

Step-by-Step Solution

Statement I Analysis:

Step 1: Create a truth table for the expression $(p \wedge \sim q) \wedge (\sim p \wedge q)$.

We need to consider all possible combinations of truth values for $p$ and $q$.

Step 2: Determine the truth values of $\sim p$ and $\sim q$.

If $p$ is true, then $\sim p$ is false, and vice-versa. Similarly, if $q$ is true, then $\sim q$ is false, and vice-versa.

Step 3: Determine the truth values of $p \wedge \sim q$.

$p \wedge \sim q$ is true only when both $p$ and $\sim q$ are true.

Step 4: Determine the truth values of $\sim p \wedge q$.

$\sim p \wedge q$ is true only when both $\sim p$ and $q$ are true.

Step 5: Determine the truth values of $(p \wedge \sim q) \wedge (\sim p \wedge q)$.

$(p \wedge \sim q) \wedge (\sim p \wedge q)$ is true only when both $p \wedge \sim q$ and $\sim p \wedge q$ are true.

The truth table is as follows:

Step 6: Analyze the final column.

Since the last column is always false, the expression $(p \wedge \sim q) \wedge (\sim p \wedge q)$ is a fallacy. Therefore, Statement I is true.

Statement II Analysis:

Step 1: Create a truth table for the expression $(p \to q) \leftrightarrow (\sim q \to \sim p)$.

Step 2: Determine the truth values of $\sim p$ and $\sim q$.

Step 3: Determine the truth values of $p \to q$.

$p \to q$ is false only when $p$ is true and $q$ is false. Otherwise, it is true. Equivalently, $p \to q \equiv \sim p \vee q$.

Step 4: Determine the truth values of $\sim q \to \sim p$.

$\sim q \to \sim p$ is false only when $\sim q$ is true and $\sim p$ is false (i.e., $q$ is false and $p$ is true). Otherwise, it is true. Equivalently, $\sim q \to \sim p \equiv q \vee \sim p$.

Step 5: Determine the truth values of $(p \to q) \leftrightarrow (\sim q \to \sim p)$.

$(p \to q) \leftrightarrow (\sim q \to \sim p)$ is true when both $p \to q$ and $\sim q \to \sim p$ have the same truth value. It is false when they have different truth values.

The truth table is as follows:

Step 6: Analyze the final column.

Since the last column is always true, the expression $(p \to q) \leftrightarrow (\sim q \to \sim p)$ is a tautology. Therefore, Statement II is true.

Step 7: Determine if Statement II is a correct explanation for Statement I.

Statement II is about the equivalence of an implication and its contrapositive. Statement I is about a compound statement that is always false. There is no direct relationship between the two. Therefore, Statement II is not a correct explanation for Statement I.

Common Mistakes & Tips

• Be careful with the order of operations when evaluating logical expressions. Parentheses are important.
• Remember the truth tables for all the logical connectives.
• When proving a statement is a tautology or fallacy, it is usually sufficient to create a truth table.
• $p \to q$ is only false when $p$ is true and $q$ is false.

Summary

Statement I is a fallacy, and Statement II is a tautology. Statement II does not explain Statement I. Therefore, Statement I is True; Statement II is True; Statement II is not a correct explanation for Statement I.

Final Answer

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