Consider the following statements :
A : Rishi is a judge
B : Rishi is honest
C Rioshi is not arrogant
The negation of the statement "if Rishi is a judge and he is not arrogant , then he is honest " is

Options

A. `B to (A vvC)`
B. `(~B )^^(A ^^C)`
C. `B to ((~A)V(~C))`
D. `B to (A ^^C)`

Answer

To find the negation of the statement "If Rishi is a judge and he is not arrogant, then he is honest," we can follow these steps:

### Step 1: Identify the components of the statement
Let:
- \( P \): Rishi is a judge.
- \( Q \): Rishi is honest.
- \( R \): Rishi is not arrogant.

The original statement can be expressed as:
\[ P \land R \implies Q \]
This means "If Rishi is a judge and he is not arrogant, then he is honest."

### Step 2: Rewrite the implication
The implication \( P \land R \implies Q \) can be rewritten using logical equivalences. The implication \( A \implies B \) is equivalent to \( \neg A \lor B \). Therefore:
\[ P \land R \implies Q \equiv \neg (P \land R) \lor Q \]

### Step 3: Find the negation of the statement
To find the negation of the statement \( P \land R \implies Q \), we negate the entire expression:
\[ \neg (\neg (P \land R) \lor Q) \]

Using De Morgan's laws, this can be simplified:
\[ \neg (\neg (P \land R)) \land \neg Q \]
This simplifies to:
\[ (P \land R) \land \neg Q \]

### Step 4: Express the negation in words
The negation of the original statement is:
"Rishi is a judge and he is not arrogant, and he is not honest."

### Final Answer
The negation of the statement "If Rishi is a judge and he is not arrogant, then he is honest" is:
"Rishi is a judge and he is not arrogant, and he is not honest."

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