Question

“If we control population growth, then we prosper”. Negative of this proposition is:
A). If we do not control population growth, we prosper
B). If we control population growth, we do not prosper
C). We control population growth and we do not prosper
D). If we do not control population growth, we do not prosper

Answer 1

First focus on the point that negation of a statement is a statement which can never be true simultaneously with the parent statement and its sub statement and its negative are mutually exclusive, i.e., one of the statements out of the two will be always true and the other statement will be always false.

Complete step-by-step solution:
Before moving to the options, let us discuss the meaning of different symbols used in the Boolean expression. See the symbols that we see in the Boolean expressions are called logic symbols, and this includes:
$\wedge$ - represents the AND logical operation.
$\vee$ - represents the OR logical operation.
$\sim$ - represents the NOT logical operation.
$\to$ - represents the IF-THEN logical operation.
The statement given to us is “If we control population growth, then we prosper”.
It contains the IF-THEN proposition.
Let P = we control population growth and Q = we prosper
The statement will be $P \to Q$.
And we know that $P \to Q$ is equivalent to $\sim P \vee Q$.
Then, the negation of $P \to Q$ will be $\sim \left( { \sim P \vee Q} \right)$.
According to De-Morgan’s Law,
$\sim \left( {A \vee B} \right) \Leftrightarrow \left( { \sim A} \right) \wedge \left( { \sim B} \right)$
Replace $A$ with $\sim P$ and $B$ with $Q$,
$\Rightarrow \sim \left( { \sim P \vee Q} \right) \Leftrightarrow \left( { \sim \left( { \sim P} \right)} \right) \wedge \left( { \sim Q} \right)$
We know that,
$\sim \left( { \sim A} \right) = A$
Use this property in the above expression,
$\Rightarrow \sim \left( { \sim P \vee Q} \right) \Leftrightarrow P \wedge \left( { \sim Q} \right)$
Thus, the negation of the statement is We control population growth and we do not prosper.
Hence, option (C) is the correct answer.

Note: Be very careful, as in such questions the play of words is very complicated. Immediately after reading the question, you will think an answer is an option (D) “We control population growth and we do not prosper”, but option (D) is the negative form of the given statement, not the negation. Always remember, two statements are the negation of each other if they are mutually exclusive and have no overlapping cases.