Question

The contrapositive of the statement “If the weather is fine then my friends will come and we will go for a picnic” is
a) The weather is fine but my friends will not come or we do not go for a picnic
b) If my friends do not come or we do not go for picnic then weather will not be fine
c) If the weather is not fine then my friends will not come or we do go for a picnic
d) The weather is not fine but my friends will come or we go for a picnic

Answer 1

Hint: We should note that in this question, we are given a conditional statement which states that if the condition weather is fine is satisfied, it implies that the result my friends will come and we will go for a picnic. To find the contrapositive, we should flip the condition and the result and then take the negation on both the condition and the resultant which will be the required answer to this question.

Complete step-by-step answer:
The given statement can be written as
The weather is fine$\Rightarrow $ friends will come and we will go for a picnic…………………………(1.1)
We know that the contrapositive of a statement $a\Rightarrow b$ is given by
 $\neg b\Rightarrow \neg a...........................(1.2)$
Where the symbol $\neg $ denotes negation i.e. if x is true, then $\neg x$ is not true and vice versa.
Also we know that the negation of a statement involving and is given by
$\neg \left( x\text{ and }y \right)=\neg x\text{ or }\neg y........................(1.3)$
Therefore, using equation (1.2) and (1.3) in (1.1), we can write the negation of the given statement as
$\neg $ (Friends will come and we will go for a picnic)$\Rightarrow $ $\neg $( The weather is fine)
$\Rightarrow $ \[\neg \] (Friends will come) or $\neg $ (we will go for a picnic)$\Rightarrow $ $\neg $( The weather is fine)
This can be also be written as
If my friends do not come or we do not go for a picnic, then the weather will not be fine which matches option (b) of the question. Thus, option (b) is the correct answer to this question.
Note: We should note that we should do negation on the whole of the resulting action in the given statement. Thus, we have to use the formula in (1.3) for converting the negation of two statements conjugated by and. However, if there were or in the original statement then we could have used the formula $\neg \left( x\text{ or }y \right)=\neg x\text{ and }\neg y$ to find the required answer.