Two Uncountable Sets Making a Infinitely Countable Set I am trying to solve the following: $A$ and $B$ are uncountable sets, what is a infinite countable result from $A \oplus B$ where $\oplus$ is the symmetric difference.
My solution:
Define $A= \mathbb Q \cup \mathbb R$ and $B= \mathbb R$
Thus, $A \oplus B = \mathbb Q$ which is countably infinite.
As you can tell, I am having a hard time justify my answers and the idea of what symmetric difference does with certain steps. Any help is greatly appreciated.

Remember that $A\oplus B=(A\cup B)\setminus(A\cap B)$.
If you want $A$ and $B$ to be uncountable, but $A\oplus B$ to be countable, that means that $A\cup B$ is "almost" $A\cap B$. But if $A=B$, we get $A\oplus B=\varnothing$, which is not infinite enough.
What you can do is reverse engineer this. Start with $C=A\oplus B$, which is countably infinite, for example $\Bbb N$, and then pick your favourite uncountable set $X$, and let $A=X$ and $B=X\cup C$. Or, split $C$ into two parts, $C_0$ and $C_1$, and let $A=C_0\cup X$ and $B=C_1\cup X$.
