If p : A man is happy and
q : A man is rich
Then, the statement ''If a man is not happy, then he is not rich'' is written as

Options

A. `~p rarr ~q`
B. `~ q rarr p`
C. `~ q rarr ~p`
D. `q rarr ~ p`

Answer

To solve the problem, we need to express the statement "If a man is not happy, then he is not rich" using the given propositions:

Let:
- \( p \): A man is happy
- \( q \): A man is rich

We need to write the statement "If a man is not happy, then he is not rich" in logical notation.

### Step-by-Step Solution:

1. **Identify the negations**:
- The negation of \( p \) (A man is happy) is \( \neg p \) (A man is not happy).
- The negation of \( q \) (A man is rich) is \( \neg q \) (A man is not rich).

2. **Formulate the conditional statement**:
- The statement "If a man is not happy, then he is not rich" can be expressed in logical terms as:
\[ \neg p \implies \neg q \]
This reads as "If \( \neg p \) (not happy), then \( \neg q \) (not rich)".

3. **Conclusion**:
- Therefore, the logical representation of the statement "If a man is not happy, then he is not rich" is:
\[ \neg p \implies \neg q \]

### Final Answer:
The statement "If a man is not happy, then he is not rich" is written as \( \neg p \implies \neg q \).

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