[Math] Cardinality of the union of two infinite set


Suppose that $A$ and $B$ are two infinite sets and $|A|<|B|$. The question is that how to prove that $|A∪B|=|B|$. The proof is related to the Axiom of Choice.

Best Answer

$|A\cup B| \leq |A| + |B| \leq |B| + |B| = |B|$ since $|B|$ is of infinite cardinality and $2\cdot \aleph_k = \aleph _k$ for whichever cardinal $|B|$ happens to be.

Then also $|B|\leq |A\cup B|$ by subadditivity.

Hence $|A\cup B| = |B|$ when $B$ is an infinite set and $|A|\leq |B|$

I don't think you need to use the axiom of choice for the proof, though perhaps I have naive notions on how set operations work on higher sized sets.

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