Exercise over numerable sets Let $A$ be a infinite set and let $D\subseteq A$ be a numerable set so that $A-D$ in infinite. Prove that $(A-D)\sim A$.

You don't say what $A$ is a subset of, so I'll prove this in a general setting. I'll also assume you mean countable by "numerable".

If $A$ is countable then the result is clear: $A - D \subset A$ is an infinite subset of a countable set, so it is also countable.

To prove the result for uncountable $A$, assume $\left|A - D\right| < \left|A\right|$. The union $D \cup (A - D)$ is equal to $A$, but its cardinality is $\left|A - D\right|$ because $D$ is countable and $A - D$ is infinite. This is a contradiction. We conclude that $(A - D) \sim A$.