Key Concepts and Formulas

• Implication ($p \Rightarrow q$): This is false only when $p$ is true and $q$ is false. Otherwise, it is true.
• Negation ($\sim p$): This is true when $p$ is false, and false when $p$ is true.
• Conjunction ($p \wedge q$): This is true only when both $p$ and $q$ are true.
• Disjunction ($p \vee q$): This is true when either $p$ or $q$ or both are true. It is false only when both $p$ and $q$ are false.
• Tautology: A statement that is always true, regardless of the truth values of its components.
• Contradiction: A statement that is always false, regardless of the truth values of its components.

Step-by-Step Solution

Step 1: Analyze Statement S1: $(p \Rightarrow q) \vee((\sim p) \wedge q)$

We need to determine if the given statement is a tautology. We'll construct a truth table to evaluate the statement for all possible truth values of $p$ and $q$.

Step 2: Construct the Truth Table for S1

$$\begin{array}{|c|c|c|c|c|c|} \hline p & q & \sim p & \sim p \wedge q & p \Rightarrow q & (p \Rightarrow q) \vee (\sim p \wedge q) \\ \hline T & T & F & F & T & T \\ \hline T & F & F & F & F & F \\ \hline F & T & T & T & T & T \\ \hline F & F & T & F & T & T \\ \hline \end{array}$$

• Column 1 & 2 (p & q): All possible combinations of truth values for $p$ and $q$.
• Column 3 ($\sim p$): Negation of $p$. If $p$ is true, $\sim p$ is false, and vice versa.
• Column 4 ($\sim p \wedge q$): Conjunction of $\sim p$ and $q$. It is true only when both $\sim p$ and $q$ are true.
• Column 5 ($p \Rightarrow q$): Implication of $p$ to $q$. It is false only when $p$ is true and $q$ is false.
• Column 6 ($(p \Rightarrow q) \vee (\sim p \wedge q)$): Disjunction of $(p \Rightarrow q)$ and $(\sim p \wedge q)$. It is true if either $(p \Rightarrow q)$ or $(\sim p \wedge q)$ or both are true.

Step 3: Determine if S1 is a Tautology

From the last column of the truth table, we see that the statement $(p \Rightarrow q) \vee (\sim p \wedge q)$ is not always true. It is false when $p$ is true and $q$ is false. Therefore, S1 is not a tautology.

Step 4: Analyze Statement S2: $(q \Rightarrow p) \Rightarrow((\sim p) \wedge q)$

We need to determine if the given statement is a contradiction. We'll construct a truth table to evaluate the statement for all possible truth values of $p$ and $q$.

Step 5: Construct the Truth Table for S2

$$\begin{array}{|c|c|c|c|c|c|} \hline p & q & \sim p & q \Rightarrow p & (\sim p) \wedge q & (q \Rightarrow p) \Rightarrow ((\sim p) \wedge q) \\ \hline T & T & F & T & F & F \\ \hline T & F & F & T & F & F \\ \hline F & T & T & F & T & T \\ \hline F & F & T & T & F & F \\ \hline \end{array}$$

• Column 1 & 2 (p & q): All possible combinations of truth values for $p$ and $q$.
• Column 3 ($\sim p$): Negation of $p$. If $p$ is true, $\sim p$ is false, and vice versa.
• Column 4 ($q \Rightarrow p$): Implication of $q$ to $p$. It is false only when $q$ is true and $p$ is false.
• Column 5 $((\sim p) \wedge q)$: Conjunction of $\sim p$ and $q$. It is true only when both $\sim p$ and $q$ are true.
• Column 6 $((q \Rightarrow p) \Rightarrow ((\sim p) \wedge q))$: Implication of $(q \Rightarrow p)$ to $((\sim p) \wedge q)$. It is false only when $(q \Rightarrow p)$ is true and $((\sim p) \wedge q)$ is false.

Step 6: Determine if S2 is a Contradiction

From the last column of the truth table, we see that the statement $(q \Rightarrow p) \Rightarrow ((\sim p) \wedge q)$ is not always false. It is true when $p$ is false and $q$ is true. Therefore, S2 is not a contradiction.

Common Mistakes & Tips

• Be careful with the truth table for implication. $p \Rightarrow q$ is only false when $p$ is true and $q$ is false.
• Remember the definitions of tautology and contradiction.
• It is very helpful to create a truth table for each statement separately.

Summary

We analyzed the two statements S1 and S2 using truth tables. We found that S1 is not a tautology because it is false when $p$ is true and $q$ is false. We also found that S2 is not a contradiction because it is true when $p$ is false and $q$ is true. Therefore, neither S1 nor S2 is true.

Final Answer

The final answer is \boxed{neither (S1) and (S2) is True}, which corresponds to option (A).