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

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Take three sets $A$, $B$, and $C$ such that $A$ and $B$ are both uncountable and disjoint and $C$ is countably infinite. Then $A\cup C$ and $B\cup C$ give you an example. –  Michael Greinecker Feb 23 '12 at 13:26
    
this is the best answer!!! –  manuzhang Feb 23 '12 at 14:00

3 Answers 3

up vote 2 down vote accepted

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

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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.

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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.

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You may learn to write answers in $\TeX$. It is neat and looks great. To see, how this is typeset, there are many manuals all around the internet. But, note that you click on the time stamp to see the edit :-) –  user21436 Feb 24 '12 at 9:56

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