Statement-1: The statement $$A \rightarrow (B \rightarrow A)$$ is equivalent to $$A \rightarrow (A \vee B)$$.
Statement-2: The statement $$\sim [(A \wedge B) \rightarrow (\sim A \vee B)]$$ is a Tautology.

We are given two statements and need to evaluate their truth values and the relationship between them. The options are based on whether each statement is true or false and if Statement-2 explains Statement-1.

First, we address Statement-1: The statement $$A \rightarrow (B \rightarrow A)$$ is equivalent to $$A \rightarrow (A \vee B)$$. Two statements are equivalent if they have the same truth value for all possible truth values of their variables. We will use truth tables to verify this.

Construct a truth table for both expressions. Since there are two variables, A and B, there are four combinations of truth values.

For $$A \rightarrow (B \rightarrow A)$$:

• Implication $$P \rightarrow Q$$ is false only when P is true and Q is false; otherwise, it is true.
• $$B \rightarrow A$$ is equivalent to $$\neg B \vee A$$.

For $$A \rightarrow (A \vee B)$$:

• $$A \vee B$$ is false only when both A and B are false; otherwise, it is true.

The truth table is as follows:

In all rows, both expressions evaluate to true. Therefore, $$A \rightarrow (B \rightarrow A)$$ and $$A \rightarrow (A \vee B)$$ are both tautologies and hence equivalent. So, Statement-1 is true.

Now, we address Statement-2: The statement $$\sim [(A \wedge B) \rightarrow (\sim A \vee B)]$$ is a tautology. A tautology is a statement that is always true, regardless of the truth values of its variables.

We simplify the given expression step by step:

• Recall that $$P \rightarrow Q$$ is equivalent to $$\neg P \vee Q$$.
• So, $$(A \wedge B) \rightarrow (\sim A \vee B)$$ is equivalent to $$\neg (A \wedge B) \vee (\sim A \vee B)$$.
• The negation of the implication is $$\sim [(A \wedge B) \rightarrow (\sim A \vee B)] = \sim [\neg (A \wedge B) \vee (\sim A \vee B)]$$.
• By De Morgan's law, $$\sim [\neg (A \wedge B) \vee (\sim A \vee B)] = (A \wedge B) \wedge \sim (\sim A \vee B)$$.
• Now, $$\sim (\sim A \vee B) = \sim (\sim A) \wedge \sim B = A \wedge \sim B$$ (by De Morgan's law).
• So, the expression becomes $$(A \wedge B) \wedge (A \wedge \sim B)$$.
• Simplify: $$(A \wedge B) \wedge (A \wedge \sim B) = A \wedge A \wedge B \wedge \sim B = A \wedge (B \wedge \sim B) = A \wedge \text{false} = \text{false}$$.

The expression simplifies to false, which is a contradiction (always false). Therefore, it cannot be a tautology. So, Statement-2 is false.

We can also verify with a truth table:

In every row, the expression $$\sim [(A \wedge B) \rightarrow (\sim A \vee B)]$$ is false. Hence, it is a contradiction, not a tautology. Thus, Statement-2 is false.

Now, evaluating the options:

• Option A: Statement-1 false; Statement-2 true → Incorrect, as Statement-1 is true and Statement-2 is false.
• Option B: Both true; Statement-2 not correct explanation → Incorrect, as Statement-2 is false.
• Option C: Statement-1 true; Statement-2 false → Correct.
• Option D: Both true; Statement-2 correct explanation → Incorrect, as Statement-2 is false.

Hence, the correct answer is Option C.