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Let $X, Y$ be two sets. We define the equality of cardinality as follows: $|X| = |Y|$ if and only if there exists bijection between $X$ and $Y$. And, $|X| \leq |Y|$ if and only if there exists one-to-one function between $X$ and $Y$. Finally, $|X| < |Y|$ if and only if $|X| \leq |Y|$ and $|X| \neq |Y|$.

Prove that $<$ is linear.

Why is the ordering follows trichotomy? The book states "It follows from the axiom of choice that it must be linear".

Duck Gia
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    You might want to look up equivalences to axiom of choice, I believe there is one regarding orders. – LGu Dec 14 '24 at 05:12
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    Obligatory linguistic note: semantically, "between $X$ and $Y$" is symmetric, so you shouldn't use that for a relation that may not be symmetric. "One-to-one function from $X$ to $Y$" is better. – Brian Moehring Dec 14 '24 at 05:23
  • You can assume AOC, then use Zorn's lemma to prove every two sets are comparable. See here https://math.stackexchange.com/questions/268942/for-any-two-sets-a-b-a-leqb-or-b-leqa – Angae MT Dec 14 '24 at 05:52
  • @AngaeMT My book did not mention Zorn's. There should exist an alternative. – Duck Gia Dec 14 '24 at 06:47
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    Then maybe this link can help you https://math.stackexchange.com/questions/421638/proving-a-le-b-vee-b-le-a-for-sets-a-and-b?noredirect=1&lq=1 – Angae MT Dec 14 '24 at 17:06

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