Question

The negative of the statement “ He is rich and happy “ is given as
A) He is not rich and not happy
B) He is not rich or not happy
C) He is rich and happy
D) He is not rich and happy

Answer 1

Hint:
By splitting the given statements into two simple statements p and q. We have the given statement to be $p \wedge q$.By using the property $\neg \left( {p \wedge q} \right) = \neg p \vee \neg q$ we get the negative of the given sentence.

Complete step by step solution:
The given statement is “ He is rich and happy”
Let's split this into two statements p and q
Let p : he is rich
And q: he is happy
So our given statement is $p \wedge q$
This can be read as p and q
That is , he is rich and happy
Now we need to find the negation of $p \wedge q$
$ \Rightarrow \neg \left( {p \wedge q} \right)$
We know that $ \Rightarrow \neg \left( {p \wedge q} \right) = \neg p \vee \neg q$
This can be read as negation of p or negation of q
Hence the negative of the given statement is he is not rich or not happy.

The correct option is B.

Note:
The property used above we have
\[ \Rightarrow \neg \left( {p \vee q} \right) = \neg p \wedge \neg q\]
That is the negation of p or q is negation of p and negation of q.
Mathematical Reasoning gives us the thought process for logical thinking, like how to think logically. In higher studies of mathematics there is a separate area for this which is discrete mathematics.