Cardinality of any two sets is comparable: proof using Zorn's lemma

Asked Dec 27 '14 at 19:50

Active Dec 27 '14 at 20:02

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Definition 1. Let $A$ and $B$ be sets. $A$ has lesser or equal cardinality to $B$ iff exists a injection from $A$ to $B$.

I tried to prove:

Let $A$ and $B$ be two non-empty sets such that $A$ does not have lesser or equal cardinality to $B$. Using the Zorn's lemma, prove that $B$ has lesser or equal cardinality to $A$.

I proved this exercise using the axiom of choice, but I could not find a relation to use the Zorn's lemma.

The book gives a hint: for every subset $X\subseteq B$, let $P(X)$ denote the property that there exists an injective map from $X$ to $A$.

I thinking about to define the $\{X\subseteq B:P(X)\}$ and show that $B\to A$ is the maximal element.

edited Dec 27 '14 at 20:02

asked Dec 27 '14 at 19:50

Cristhian Gz

2,559

There were actually several threads on the topic in the past. Let me find one. – Asaf Karagila Dec 27 '14 at 19:53
I was searching, but I couldn't find anyone. – Cristhian Gz Dec 27 '14 at 19:54
http://math.stackexchange.com/questions/421638/proving-a-le-b-vee-b-le-a-for-sets-a-and-b seems like a reasonable candidate. – Asaf Karagila Dec 27 '14 at 19:54
If I were to write a hint, it would probably be some variant of what I wrote in my answer there. If you have more specific problems, it'd be great if you can add them to your question. – Asaf Karagila Dec 27 '14 at 19:55
Great! It's good for me. A lot of thanks. :) – Cristhian Gz Dec 27 '14 at 19:55
Should I close this as a duplicate, then? – Asaf Karagila Dec 27 '14 at 19:56
I don't know the procedure. What is the best option: close as duplicate or delete the question? – Cristhian Gz Dec 27 '14 at 19:57
I'll close, you can delete it later if you like. :-) – Asaf Karagila Dec 27 '14 at 19:58
OK. Thanks. :-) – Cristhian Gz Dec 27 '14 at 19:59

Cardinality of any two sets is comparable: proof using Zorn's lemma

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