Question

Negation of the compound statement ‘if the examination is difficult, then I shall pass if I study hard’ is ___.

• A. The examination is difficult and I study hard and I shall pass
• B. The examination is difficult and I study hard and I shall not pass
• C. The examination is not difficult and I study hard and I shall not pass
• D. The examination is difficult and I do not study hard and I shall pass

Answer 1

The correct option is B

The examination is difficult and I study hard and I shall not pass

$\sim (p\Rightarrow q)=p\wedge \sim q$
p: The examination is difficult
q: I study hard
r: I shall not pass

The given statement is $p\Rightarrow (q\Rightarrow r)$. Its negation is
$\sim p\Rightarrow (q\Rightarrow r)=p\wedge \sim (q\Rightarrow r)=p\wedge (q\wedge \sim r)$

The negation of the given statement is $p\wedge (q\wedge \sim r)$ i.e. 'The examination is difficult and I study hard and I shall not pass'.