How would the intersection of two uncountable sets be finite?

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This is a problem from Discrete Mathematics and its Applications

Here is my book's definition on countable

and definition of having the same cardinality

The only example that my book gave of uncountable set was the set of real numbers. I understand that because if you try listing out all of the members of the set, you would keep going on and on - 1, 1.01, 1.001, etc...... But the intersection of the set of real numbers and itself is the set of real numbers is uncountable as well... Is there another uncountable set that you could use to prove this?

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3 Answers

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Here's another idea,

$$(-\infty,0]\cap[0,\infty)=\{0\}$$

What can you say about the cardinality of $(-\infty,0]$ and $[0,\infty)$?

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Here's one idea: suppose $C$ and $D$ are disjoint uncountable sets, and $E$ is finite. Consider $A=C \cup E$ and $B=D \cup E$. Then $A \cap B=E$ is finite. Can you come up with two disjoint uncountable sets to set this up?

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Can you give an example of an uncountable set $B$ that is disjoint from the uncountable set $A= [0,1)$ ??

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