Key Concepts and Formulas

• Implication: $A \Rightarrow B$ is equivalent to $\sim A \vee B$.
• Truth Table for AND: $p \wedge q$ is true only when both $p$ and $q$ are true.
• Truth Table for OR: $p \vee q$ is true when at least one of $p$ or $q$ is true.
• Truth Table for NOT: $\sim p$ is the negation of $p$.

Step-by-Step Solution

Step 1: Construct the truth table for the given expression $(p \wedge \sim q) \Rightarrow (q \vee \sim p)$.

We will evaluate the expression for all possible truth values of $p$ and $q$. This is a standard method for simplifying Boolean expressions.

Step 2: Construct the truth table for the option (A) $q \Rightarrow p$.

Using the implication rule $A \Rightarrow B \equiv \sim A \vee B$, we have $q \Rightarrow p \equiv \sim q \vee p$.

Step 3: Construct the truth table for the option (B) $p \Rightarrow q$.

Using the implication rule $A \Rightarrow B \equiv \sim A \vee B$, we have $p \Rightarrow q \equiv \sim p \vee q$.

Step 4: Construct the truth table for the option (C) $\sim q \Rightarrow p$.

Using the implication rule $A \Rightarrow B \equiv \sim A \vee B$, we have $\sim q \Rightarrow p \equiv \sim (\sim q) \vee p \equiv q \vee p$.

Step 5: Construct the truth table for the option (D) $p \Rightarrow \sim q$.

Using the implication rule $A \Rightarrow B \equiv \sim A \vee B$, we have $p \Rightarrow \sim q \equiv \sim p \vee \sim q$.

Step 6: Compare the truth tables.

From Step 1, the truth table for $(p \wedge \sim q) \Rightarrow (q \vee \sim p)$ is T, F, T, T.

From Step 2, the truth table for $q \Rightarrow p$ is T, T, F, T.

From Step 3, the truth table for $p \Rightarrow q$ is T, F, T, T.

From Step 4, the truth table for $\sim q \Rightarrow p$ is T, T, T, F.

From Step 5, the truth table for $p \Rightarrow \sim q$ is F, T, T, T.

Comparing the truth tables, we see that $(p \wedge \sim q) \Rightarrow (q \vee \sim p)$ is equivalent to $p \Rightarrow q$.

Common Mistakes & Tips

• Remember that $A \Rightarrow B$ is equivalent to $\sim A \vee B$. This is crucial for simplifying implications.
• Be careful with the order of operations and negations when constructing the truth tables.
• Double check the truth values in your truth tables to avoid errors.

Summary

We constructed truth tables for the given expression and all the options. By comparing the truth tables, we found that the expression $(p \wedge \sim q) \Rightarrow (q \vee \sim p)$ is equivalent to $p \Rightarrow q$.

Final Answer

The final answer is \boxed{p \Rightarrow q}, which corresponds to option (B).