Question 7

Let $\mathbb{A}$ and $\mathbb{B}$ be sets of reals, such that between any two distinct elements of $\mathbb{A}$ there is an element of $\mathbb{B}$ and vice-versa, between any two distinct elements of $\mathbb{B}$ there is an element of $\mathbb{A}$.

• Prove that if $\mathbb{A}$ is finite, then so is $\mathbb{B}$ and the the number of elements of the set $\mathbb{A}$ differs from the number of elements of the set $\mathbb{B}$ by at most one.
• Give an example (with proof) where $\mathbb{A}$ is countable and $\mathbb{B}$ is uncountable and $\mathbb{A} \cap \mathbb{B} = \phi$.
• Suppose that $\mathbb{A}$ is infinite and bounded.
  Prove that $\mathbb{B}$ is also infinite and bounded.
  Suppose also that $\sup(\mathbb{A}) \notin \mathbb{A}$ and $\sup(\mathbb{B}) \notin \mathbb{B}$.
  Prove that $\sup(\mathbb{A}) = \sup(\mathbb{B})$.