Question

Consider the following statements:
\( A \): Rishi is a judge.
\( B \): Rishi is honest.
\( C \): Rishi is not arrogant.
The negation of the statement "If Rishi is a judge and he is not arrogant, then he is honest" is:

• A. \( B \rightarrow (A \vee C) \)
• B. \( (\neg B) \wedge (A \wedge C) \)
• C. \( B \rightarrow ((\neg A) \vee (\neg C)) \)
• D. \( B \rightarrow (A \wedge C) \)

Hint:

Negation of an implication \( P \rightarrow Q \) is always \( P \wedge \neg Q \).

Answer 1

The Correct Option is B

The given statement is: \[ (A \wedge C) \rightarrow B \] The negation of an implication \( P \rightarrow Q \) is given by: \[ \neg (P \rightarrow Q) \equiv P \wedge \neg Q \] Thus: \[ \neg [(A \wedge C) \rightarrow B] \equiv (A \wedge C) \wedge \neg B \] which matches option (B).

Answer 2

Step 1: Understanding the given statement
The original statement is: "If Rishi is a judge and he is not arrogant, then he is honest." We can represent this statement logically as follows: - \( A \): Rishi is a judge, - \( B \): Rishi is honest, - \( C \): Rishi is not arrogant.

The statement can be written as: \[ (A \wedge C) \Rightarrow B \] This means "If Rishi is a judge and he is not arrogant, then he is honest."

Step 2: Finding the negation
To find the negation of the statement \( (A \wedge C) \Rightarrow B \), we use the logical equivalence that: \[ \neg (P \Rightarrow Q) \equiv P \wedge \neg Q \] So the negation of \( (A \wedge C) \Rightarrow B \) will be: \[ (A \wedge C) \wedge \neg B \] This means "Rishi is a judge and he is not arrogant, and Rishi is not honest."

Step 3: Final Answer
The correct negation of the statement is:

\( (\neg B) \wedge (A \wedge C) \)