Question

Consider the following statements: 
P : I have fever 
Q: I will not take medicine 
R : I will take rest 
The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:

• A. $(P \vee Q) \wedge((\sim P) \vee R)$
• B. $(P \vee \sim Q) \wedge(P \vee \sim R)$
• C. $((\sim P) \vee \sim Q) \wedge((\sim P) \vee \sim R)$
• D. $((\sim P) \vee \sim Q)\wedge((\sim P) \vee R)$

Answer 1

The Correct Option is D

To solve this problem, we need to rewrite the given conditional statement into a logical expression and then find its equivalent form among the options provided. Let's analyze the statement and break it down step-by-step:

The given statement is: "If I have fever, then I will take medicine and I will take rest."

This can be represented in logical symbols as:

• \(P\): I have fever
• \(Q\): I will not take medicine
• \(R\): I will take rest

The statement "If I have fever, then I will take medicine and I will take rest" can be rewritten using logical implications as:

\(P \rightarrow (\sim Q \land R)\)

Using the logical equivalence of implications, we know that \(A \rightarrow B\) is equivalent to \(\sim A \lor B\). Therefore, the above expression can be rewritten as:

\(\sim P \lor (\sim Q \land R)\)

Using distribution of logical operators:

\((\sim P \lor \sim Q) \land (\sim P \lor R)\)

This is the equivalent expression for the original statement.

Now, let's compare this with the provided options to find the correct match:

• \((P \vee Q) \wedge((\sim P) \vee R)\) - This does not match.
• \((P \vee \sim Q) \wedge(P \vee \sim R)\) - This does not match.
• \(((\sim P) \vee \sim Q) \wedge((\sim P) \vee \sim R)\) - This does not match because of the second component.
• \(((\sim P) \vee \sim Q)\wedge((\sim P) \vee R)\) - This matches!

Thus, the correct answer is:

\(((\sim P) \vee \sim Q) \wedge ((\sim P) \vee R)\)

This confirms that the logical expression we derived correctly matches the fourth option. Therefore, this is the equivalent logical statement for the given conditional statement.