A problem in Elements of the Theory of Computation
Examples to show intersection of two uncountable sets can be countably infinite
Examples to show intersection of two uncountable sets can be countably infinite
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elementary-set-theory
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0this is the best answer!!! – 2012-02-23
3 Answers
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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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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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0You 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 :-) – 2012-02-24
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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.