Question

# 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 corresponds to which statement? \begin{align} & \left[ a \right]\sim p \to \sim q \\ & \left[ b \right]\sim q\to p \\ & \left[ c \right]\sim q\to \sim p \\ & \left[ d \right]q\to \sim p \\ \end{align}

Hint:We shall solve the question option wise. Write the statement form of each of the options and check which of the statement means the same as the target statement and hence find which of the options is correct. The negation of the statement is the statement(denoted as ~) obtained by adding not to the original statement as is correct when the original statement is incorrect. Similarly, the implication statement $p\to q$ is read as if p then q and is true if p is false or if p is true, and q is true.

The rule for converting p to ~p:
Negate the statement by adding not
P: A man is happy
Hence we have
~P: A man is not happy
q: A main is rich
Hence, we have
~q: A man is not rich
Rule for converting $p\to q$
Add the if-then statement, i.e. if p then q
Hence, we have
$\sim p\to \sim q:$ If a man is not happy, then he is not rich
$\sim q\to p$ : If a man is not rich, then he is happy
$\sim q\to \sim p$ : If a man is not rich, then he is not happy
$q\to \sim p$ : If a man is rich, then he is not happy
Hence option [a] corresponds to the target statement.
Hence option [a] is correct.
Note: Alternatively, we can backtrack the target statement into p and q statements as follows
Let r: Man is not happy
s: Man is not rich.
Let t be the target statement
Hence, we have
$t=r\to s$
But, we have r = ~p and s = ~q
Hence, we have
$t=\sim p\to \sim q$ , which is the same as obtained above.
Hence option [a] is correct.