Prove $ |A \cup B| = |[0,1]| \Rightarrow |A|=|[0,1]| \bigvee |B|=|[0,1]| $

Question

I've tried the argument by contradiction, but did not succeed.

Intuitively I understand that union of less-than-continuum sets cannot equal to $|\mathbb{R}|$. I'm curious what a formal proof could be.

Answer 1

There is no naive proof.

More specifically, any proof is going to have to rely on the following property of infinite cardinals: $$\kappa+\lambda=\max\{\kappa,\lambda\}.$$ This property follows from the axiom of choice (and as stated here is in fact equivalent to the axiom of choice). So some nontrivial work is needed.

Why is it needed? Well, it turns out that it is consistent without the axiom of choice that there are $A,B\subseteq[0,1]$ such that $A\cup B=[0,1]$, but both $A$ and $B$ have cardinality less than the continuum. This is due to G. P. Monro.

Monro, G.P., Decomposable cardinals, Fundam. Math. 80, 101-104 (1973). ZBL0272.02085.

The most you can prove naively without choice, is that if $A$ is countable, then $|B|=|[0,1]|$. (You can say more, of course, but that would require getting your hands dirty in technical concepts of set theory.)

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Assuming the aforementioned property of infinite cardinals, the proof is nearly trivial. $|A\cup B|=\max\{|A|,|B|\}$. So if that would be $|[0,1]|$, it has to be that one of $A$ or $B$ has the same cardinality.