Examples to show intersection of two uncountable sets can be countably infinite

Question

A problem in Elements of the Theory of Computation
Examples to show intersection of two uncountable sets can be countably infinite

Answer 1

You have to make up your set for example in $\mathbb{R}$ you consider, $\mathbb{R}_{+} \cup \mathbb{Q}$ and $\mathbb{R}_{-}\cup \mathbb{Q}$.

Answer 2

Consider the sets: $A=\{0\}\times\mathbb R$ and $B=\{1\}\times\mathbb R$. Both of these sets are uncountable and disjoint.

Now consider $A'=A\cup\mathbb N$ and $B'=B\cup\mathbb N$. These new sets are still uncountable and $A'\cap B'=\mathbb N$ is countable.

Furthermore, consider the intervals $(0,1)$ and $(2,3)$. Now consider $(0,1)\cup\mathbb Q$ and $(2,3)\cup\mathbb Q$. The same argument applies.

Answer 3

Let $A=(-\infty,0] \cup \{1,2,3,4,5,...\}$; $B=[0,\infty)$. Verify that $A$ and $B$ satisfy the criterion you are asking for.