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?
why does the wikipedia proof (and some of the other proofs) assume that A and B are disjoint? what does this buy us?

Unified notation

Let $A, B$ be sets. (In Tao's book we assume $A \subseteq B$ but this isn't the case in the other proofs.)

$f : A \to B$ (In Tao's book we assume that $f = ι_{A \to B}$ is the inclusion map that sends each $x \in A$ to itself.)

$g : B \to A$

$C_{0} = A ∖ g (B)$

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

$D_{0} = B ∖ f (A)$

$D_{n + 1} = (f \circ g) (D_{n})$ for all $n \geq 0$ (In Tao's book, since $f$ is just the inclusion map and $g$ maps into $A$ , we have $f \circ g = g$ which is why Tao can write $D_{n + 1} = g (D_{n})$ . Note that Tao uses "f" instead of "g" because his notation is different from everybody else's.)

Chain types

Type 1: Loop

Type 2: infinitely goes backwards without repeating

Type 3: Chain stops in A, with $g^{- 1}$ eventually undefined.

3a: Elements of $C_{0}$ ( $g^{- 1}$ immediately undefined)
3b: Elements of $⋃_{n = 1}^{\infty} C_{n}$ ( $g^{- 1}$ is defined, but eventually if you keep going $g^{- 1}$ will be undefined)

Type 4: Chain stops in B, with $f^{- 1}$ eventually undefined.

4a: Elements of $D_{0}$ ( $f^{- 1}$ immediately undefined)
4b: Elements of $⋃_{n = 1}^{\infty} D_{n}$ ( $f^{- 1}$ is defined, but eventually if you keep going $f^{- 1}$ will be undefined)

Definitions of the bijection in various books

Book of Proof: h(x) = f(x) if x is of type 3; $h (x) = g^{- 1} (x)$ if x is of type 1,2,4.
Cornell page: h(x) = f(x) if x is of type 1,2,3; $h (x) = g^{- 1} (x)$ if x is of type 4.
Tao: h(x) = f(x) if x is of type 1,2,3,4a (actually case 4a doesn't even exist in the Tao book since A is a subset of B); $h (x) = g^{- 1} (x)$ if x is of type 4b
Wikipedia: h(x) = f(x) if x is of type 3; $h (x) = g^{- 1} (x)$ if x is of type 4; h(x) can be defined either way if x is of type 1 or 2

Unified notation

Chain types

Definitions of the bijection in various books

References