.Consider the following statements:
`P:` I have fever
`Q:` I will not take medicine
`R:1` will take rest.
The statement " If I have fever, then I will take medicine and I will take rest " is equivalent to :

Options

A. `(P vvQ) vv((~P) vvR)`
B. `(P vv~Q) ^^ ((Pvv~R)`
C. `(~P) vv~Q ^^((~P)vv~R)`
D. `((~P) vv~Q) ^^((~P)vvR)`

Answer

To solve the problem, we need to analyze the given statements and translate them into logical expressions.

### Step 1: Understand the statements
We have three statements:
- \( P: \) I have fever
- \( Q: \) I will not take medicine
- \( R: \) I will take rest

The statement we need to analyze is: "If I have fever, then I will take medicine and I will take rest."

### Step 2: Translate the statement into logical form
The statement can be expressed in logical terms:
- The phrase "If I have fever" translates to \( P \).
- The phrase "I will take medicine" can be expressed as \( \neg Q \) (since \( Q \) is "I will not take medicine").
- The phrase "I will take rest" translates to \( R \).

Thus, the entire statement can be written as:
\[ P \implies (\neg Q \land R) \]

### Step 3: Apply logical equivalences
Using the implication equivalence, we can rewrite \( P \implies (\neg Q \land R) \):
\[ P \implies (\neg Q \land R) \equiv \neg P \lor (\neg Q \land R) \]

### Step 4: Use distribution
Now we can distribute the terms:
\[ \neg P \lor (\neg Q \land R) \equiv (\neg P \lor \neg Q) \land (\neg P \lor R) \]

### Step 5: Final expression
Thus, the final expression we derived is:
\[ (\neg P \lor \neg Q) \land (\neg P \lor R) \]

### Step 6: Identify the equivalent statement
Now, we can check which of the provided options matches this expression.

### Conclusion
After analyzing the options, we find that the correct equivalent statement is:
\[ \neg P \lor \neg Q \land \neg P \lor R \]