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 $\#B=\#\mathbb{Z}$.

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.

Answer 1

$(1)$ 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.