Key Concepts and Formulas

• Implication: $P \to Q \equiv \sim P \vee Q$
• Distributive Law: $A \vee (B \wedge C) \equiv (A \vee B) \wedge (A \vee C)$

Step-by-Step Solution

Step 1: Translate the given statement into symbolic form. The statement "If I have fever, then I will take medicine and I will take rest" can be written as $P \to (Q \wedge R)$. Here, $P$ is "I have fever", $Q$ is "I will take medicine", and $R$ is "I will take rest". Note that in the question, Q is "I will NOT take medicine". Thus we need to change Q to ~Q to match the question. The statement is then $P \to (\sim Q \wedge R)$.

Step 2: Apply the implication equivalence. We know that $P \to Q \equiv \sim P \vee Q$. Applying this to our statement, we get: $P \to (\sim Q \wedge R) \equiv \sim P \vee (\sim Q \wedge R)$ This step uses the implication equivalence formula to rewrite the "if-then" statement using "or" and negation.

Step 3: Apply the distributive law. We can distribute the "or" ($\vee$) over the "and" ($\wedge$) using the distributive law: $A \vee (B \wedge C) \equiv (A \vee B) \wedge (A \vee C)$. Applying this to our expression, we get: $\sim P \vee (\sim Q \wedge R) \equiv (\sim P \vee \sim Q) \wedge (\sim P \vee R)$ This step transforms the expression into a form that matches one of the answer choices.

Common Mistakes & Tips

• Remember the correct equivalence for implication: $P \to Q \equiv \sim P \vee Q$. A common mistake is to mix up the negation.
• Carefully note the given definitions of P, Q, and R. Pay attention to negations. The question statement says Q: I will not take medicine.
• The distributive law is a useful tool for manipulating logical expressions. Make sure to apply it correctly.

Summary

We started with the given statement in symbolic form, $P \to (\sim Q \wedge R)$. We then applied the implication equivalence to rewrite it as $\sim P \vee (\sim Q \wedge R)$. Finally, we used the distributive law to obtain $(\sim P \vee \sim Q) \wedge (\sim P \vee R)$. This matches option (A).

Final Answer

The final answer is $(\sim P \vee \sim Q) \wedge (\sim P \vee R)$, which corresponds to option (A). The final answer is \boxed{((\sim P) \vee \sim Q) \wedge((\sim P) \vee R)}.