Question

Negation of the compound proposition: If the examination is difficult, then I shall pass if I study hard

• A. The examination is difficult and I study hard and I shall pass
• B. The examination is difficult and I study hard but I shall not pass
• C. The examination is not difficult and I study hard and I shall pass
• D. None of these

Answer 1

The correct option is B

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

Explanation for the correct option:

Finding the negation of the compound proposition:

Let p: examination is difficult

q: I shall pass

r: I study hard

$\mathrm{p}\Rightarrow (\mathrm{r}\Rightarrow \mathrm{q})$

$~(\mathrm{r}\Rightarrow \mathrm{q})=\mathrm{r}\wedge ~\mathrm{q}$

$~(\mathrm{p}\Rightarrow (\mathrm{r}\Rightarrow \mathrm{q}\left)\right)\equiv \mathrm{p}\wedge (\mathrm{r}\wedge ~\mathrm{q})$

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

Hence, the correct answer is Option (B).