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 → (A ∨ C)
• B. (~ B) ∧ (A ∧ C)
• C. B → ((~ A) ∨ (~ C))
• D. B → (A ∧ C)

Answer 1

The Correct Option is B

 To determine the negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest," we must first express the given statement in logical terms. The original conditional statement can be expressed as:

\(A \land \lnot C \rightarrow B\)

• A: Rishi is a judge.
• B: Rishi is honest.
• C: Rishi is not arrogant.

The logical expression for "if Rishi is a judge and he is not arrogant, then he is honest" is \((A \land \lnot C) \rightarrow B\).

The negation of a conditional statement \(P \rightarrow Q\) is \(\lnot Q \land P\). Applying this negation rule, we find the negation:

\(\lnot B \land (A \land \lnot C)\)

This shows that the negation implies "Rishi is not honest, and Rishi is a judge and is not arrogant."

Simplifying the expression, we have:

\(\lnot B \land (A \land C)\)

Therefore, the correct option is (\(\lnot B\)) \(\land (A \land C)\).

Option	Expression
B → (A ∨ C)	\(B \rightarrow (A \lor C)\)
(~ B) ∧ (A ∧ C)	\(\lnot B \land (A \land C)\)
B → ((~ A) ∨ (~ C))	\(B \rightarrow (\lnot A \lor \lnot C)\)
B → (A ∧ C)	\(B \rightarrow (A \land C)\)

Option (b): \((\lnot B) \land (A \land C)\) matches our derived result, confirming it as the correct answer.