Question 3

Let $f: \mathbb{A} \rightarrow \mathbb{B}$ be an injection.

• Suppose that $\mathbb{A}$ is countably infinite.
  Prove that there is an injection from $\mathbb{B}$ to $\mathbb{A}$ if and only if $\mathbb{B}$ is countably infinite.
• Suppose instead that $\mathbb{A}$ is uncountable.
  Can $\mathbb{B}$ be countable? Explain your answer.