Consider the proposition : " if we control polulation growth, we prosper". Negative of this proposition is

Options

A. If we do not contral population growth , we prosper
B. If we control propulation, we do not prosper
C. we contral population but we do not prosper
D. we do not contral propulation but we prosper

Answer

To find the negative of the proposition "If we control population growth, we prosper," we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Proposition**:
The given proposition can be represented as "If P, then Q," where:
- P: We control population growth.
- Q: We prosper.

2. **Symbolic Representation**:
The proposition can be written in symbolic form as \( P \implies Q \).

3. **Negate the Proposition**:
The negative of the implication \( P \implies Q \) can be expressed as \( \neg (P \implies Q) \).

4. **Use Implication Identity**:
We know that \( P \implies Q \) is equivalent to \( \neg P \lor Q \). Therefore, we can rewrite the negation:
\[ \neg (P \implies Q) = \neg (\neg P \lor Q) \]

5. **Apply De Morgan’s Laws**:
According to De Morgan's laws, the negation of a disjunction is the conjunction of the negations:
\[ \neg (\neg P \lor Q) = P \land \neg Q \]

6. **Interpret the Result**:
The expression \( P \land \neg Q \) translates back into words as:
"We control population growth but do not prosper."

### Final Answer:
The negative of the proposition "If we control population growth, we prosper" is:
"We control population growth but do not prosper."