Question

Negation of the conditional, “If it rains, I shall go to school” is:
(a) It rains and I shall go to school
(b) It rains and I shall not go to school
(c) It does not rains and I shall go to school
(d) None of the above

Answer 1

To solve this statement, we should first assume the variables for both the statements as “it rains” and “I shall go to school”. Then as it is a conditional statement, so it is of the form $p\to q$ and negation of it would be $\sim \left( p\to q \right)$ also represented as $p\wedge \sim q$ where $\sim q$ is the negation of q.

Complete step-by-step solution:
We are given a conditional statement as “If it rains, I shall go to school”
Let p be the statement, “It rains”
$\Rightarrow p=\text{It rains}$
Let q be the statement which denotes “I shall go to school”
$\Rightarrow q=\text{I shall go to school}$
Converting the conditional statement “If it rains, I shall go to school” into p and q
$\Rightarrow p\to q$
And negation of the conditional statement $p\to q$ is:
$\Rightarrow \sim \left( p\to q \right)$
where $\sim r$ is the negation of r statement.
$\sim \left( p\to q \right)$ is also denoted by $p\wedge \sim q$ where $\wedge$ is the symbol of “and”.
So, the negation of $p\to q$ is $p\wedge \sim q$
The negative of “If it rains, I shall go to school” is “It rains and I shall not go to school”.
Hence, option (b) is the right answer.

Note: If the student has any confusion on how the $\sim \left( p\to q \right)$ is also denoted by $p\wedge \sim q$ we can equate truth tables of both the given as below:

p	q	$p\to q$	$\sim \left( p\to q \right)$	$\sim q$	$p\wedge \sim q$
T	T	T	F	F	F
T	F	F	T	T	T
F	T	T	F	F	F
F	F	T	F	T	F

So, we observe that the value $\sim \left( p\to q \right)$ and $p\wedge \sim q$ are equal. Hence, they both are the same.