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Suppose $A$ and $B$ are sets with $\#A=\#\mathbb{Z}$ and $\#B=\#\mathbb{Z}$.

  1. Prove that $\#(A\cup B)=\#\mathbb{Z}$.
  2. Is it necessarily true that $\#(A\cap B)=\#\mathbb{Z}$?

I'm sure that I need to use bijection to prove this, but I don't know the exact way since these are new stuff for me.

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  • $\begingroup$ Does your $Z$ mean $\mathbb{Z}$, the set of all integers? Or is $Z$ an arbitrary set? In the latter case, the statement in (a) isn't true... $\endgroup$
    – zipirovich
    Dec 20, 2016 at 23:14
  • $\begingroup$ It means the set of all integers, I just couldn't paste the letter like this here $\endgroup$
    – George S
    Dec 20, 2016 at 23:16

1 Answer 1

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$(1)$ is a well known fact that countable unions (in this case, finite) of countable sets are countable.

$(2)$ is false; consider $A$ the set of even integers and $B$ the set of odd integers.

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  • $\begingroup$ Ohh yeah it makes sense now, thank you a lot $\endgroup$
    – George S
    Dec 20, 2016 at 23:38

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