What would be the contrapositive of a statement about infima?

Question

I need some help for an assignment. I read a theorem that said

If $S$ is a non-empty and bounded subset of $\Bbb R$, then $\inf S$ is a unique value.

The positive statement would be

If $\inf S$ is not a unique value then $S$ is empty or unbounded.

Am I correct?

Answer 1

Yes, that is the contrapositive, if $S$ is understood as a subset of $\Bbb R$.

Answer 2

In general, for a conditional statement $P \Rightarrow Q$, the contrapositive is $\neg Q \Rightarrow \neg P$.

Furthermore, a conditional statement is logically equivalent to its contrapositive. You can convince yourself of this by filling out a truth table.