Consider the following statements: A : Rishi is a judge. B : Rishi...

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

Options

A $B \to \left(A ∨ C\right)$
B $\left(~ B\right) ∧ \left(A ∧ C\right)$
C $B \to \left(\left(~ A\right) ∨ \left(~ C\right)\right)$
D $B \to \left(A ∧ C\right)$

Correct answer

B. $\left(~ B\right) ∧ \left(A ∧ C\right)$

Step-by-step solution

if Rishi is a judge and he is not arrogant, then he is honest can be written as $\left(A ∧ C\right) \to B$

Now, Negation of \left(A ∧ C\right) \to B $is$ ~ \left(\left(A ∧ C\right) \to B\right) $</p><p>$ = ~ \left(~ \left(A ∧ C\right) ∨ B\right) ${Using De-Morgan's law}</p><p>$ = \left(A ∧ C\right) ∧ \left(~ B\right)$

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