Suppose $A$ and $B$ are sets with $\#A=\#\mathbb{Z}$ and $\#B=\#\mathbb{Z}$. (a) Prove $\#(A\cup B)=\#\mathbb{Z}$. (b) Is $\#(A\cap B)=\#\mathbb{Z}$?

Question

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

Prove that $\#(A\cup B)=\#\mathbb{Z}$.

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.

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... — zipirovich, Dec 20 '16 at 23:14
It means the set of all integers, I just couldn't paste the letter like this here — George S, Dec 20 '16 at 23:16

score 3 · Accepted Answer · answered Dec 20 '16 at 23:26

3

$(1)$ 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.

answered Dec 20 '16 at 23:26

Fimpellizzeri

23,126

Ohh yeah it makes sense now, thank you a lot – George S Dec 20 '16 at 23:38

Suppose $A$ and $B$ are sets with $\#A=\#\mathbb{Z}$ and $\#B=\#\mathbb{Z}$. (a) Prove $\#(A\cup B)=\#\mathbb{Z}$. (b) Is $\#(A\cap B)=\#\mathbb{Z}$?

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