User:IssaRice/Schröder–Bernstein theorem

questions:

• are the ancestor proof and the iterated partition proof actually different, or do they just look different on the surface? btw this is a very nice writeup of the ancestor proof (much clearer than wikipedia's version); HT Satira. UPDATE: these are actually basically the same proof, just written in different notation.
• what is the proof that uses axiom of choice, and how does choice simplify the proof?
• is the result a fixed point result? (if so how can we phrase it as such?) or is it just that some of the proofs makes use of the fixed point ideas?

Unified notation

$f:A\to B$

$g:B\to A$

$C_{0}=A\setminus g(B)$

$C_{n+1}=(g\circ f)(C_{n})$ for all $n\geq 0$

$D_{0}=B\setminus f(A)$

$D_{n+1}=(f\circ g)(D_{n})$ for all $n\geq 0$

References

• https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem
• https://www.cs.cornell.edu/courses/cs2800/2017sp/lectures/lec05-cantor.html
• http://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf#page=242
• https://math.stackexchange.com/questions/3323133/questions-about-the-schroeder-bernstein-theorem
• https://math.stackexchange.com/questions/1726578/understanding-a-proof-of-schr%C3%B6der-bernstein-theorem
• https://proofwiki.org/wiki/Cantor-Bernstein-Schr%C3%B6der_Theorem/Lemma
• https://proofwiki.org/wiki/Cantor-Bernstein-Schr%C3%B6der_Theorem