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After define the equality of the cardinality of two sets, I defined the following ordering in the class of the all sets: let $A,B$ be sets, we define $|A|\le|B|$ if there is a injection $f:A\to B$.

Now, I'm trying to prove that $|A|\ge |B|$ iff there is a surjection $g:A\to B$ using the definitions I made.

I know this is true intuitively, but I couldn't prove it formally using my definitions, I need help.

Another question is this construction is standard in set theory?

Thanks in advance

user42912
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1 Answers1

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Let $g : A \to B$ be a surjection, then there exists a right inverse:

Surjection iff right inverse

and this right inverse map, has a left inverse, namely $g$ itself, and so is injective since:

Injectiion iff left inverse

Call it $g^{\leftarrow}$, then $g^{\leftarrow} : B \to g^{\leftarrow}(B) \subset A$ is a bijection. Thus you have a bijection of $B$ onto a subset of $A$. What does that mean?

Now, if $|A| \geq |B|$, in particular $|B| \leq |A|$ and you have an injection $f: B \to A$. And using the above-reffered to articles, there's a left inverse of $f$, $g$ which is surjective. You do the rest. QED

Daniel Donnelly
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