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

Hint: 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.