# Math Help - [SOLVED] Cantor-Schroder-Bernstein

1. ## [SOLVED] Cantor-Schroder-Bernstein

Consider the injections $f : \mathbb{N} \rightarrow \mathbb{N} \sqcup \mathbb{N} : n \rightarrow (0, n)$ and $g : \mathbb{N} \sqcup \mathbb{N} \rightarrow \mathbb{N}$ given by
$(i, n) \rightarrow 3n + i$, where $i = 0$ or $i = 1$.
(a) From the proof of Cantor-Schroder-Bernstein compute F(n) directly for n =
0, 1, 2, 3, 4 and find $F^{-1} (0,4)$.

Is this just simply finding g o f? Like for n=2 it would be f(2) = 2 then g(0,2) = 6?

Or is it... Find x such that (g o f)(x) = 0,1,2,3,4?

OR! Is it something to do with the fact that... if i=0 then F(n) can be either 0,3,6, ... (3n + 0 basically). And if i=1 then F(n) can be either 1,4,7, ... (3n+1)..?

Since F isn't actually defined I'm not sure what it is.

AND. Since the CSB theorem is involved, there will be bijection A -> B so it makes me think that some values of n will equal 0,3,6, ... or 1,4,7 ... and the rest will have to be defined as something else...

Consider the injections $f : \mathbb{N} \rightarrow \mathbb{N} \sqcup \mathbb{N} : n \rightarrow (0, n)$ and $g : \mathbb{N} \sqcup \mathbb{N} \rightarrow \mathbb{N}$ given by
$(i, n) \rightarrow 3n + i$, where $i = 0$ or $i = 1$.
(a) From the proof of Cantor-Schroder-Bernstein compute F(n) directly for n = 0, 1, 2, 3, 4 and find $F^{-1} (0,4)$.
Do you realize that there are as many difference proofs of the Cantor-Schroder-Bernstein theorem as there are places where a proof is published?
So there is no way for us to know how your $F$ is defined.

3. Bleh im dumb.

Proof is the one with the zig-zig diagram that has A on one side, B on the other, f is an injection from A to B and B is an injection from B to A.
Then you repeatedly apply f and g to an element $a \in A$.
Now, we must be in one of two situations: either

(a) we can go up indefinitely from a or end on the A side, or,
(b) we end on the B side.

Let $A_1 = a \in A$, a is type a, $A_2 = a \in A$, a is type b. Then $A = A_1 \cup A_2$ and $A_1 \cap A_2 = \emptyset$.

From this we have a well defined map $F : A \rightarrow B$ given by...

$F(a) = f(a)$ if $a \in A_1$, $g^{-1}(a)$ if $a \in A_2$
Taken from the notes!

Indeed I probably shoulda noticed this bit first...

So, Does this mean that F(0), F(1), etc = f(0), f(1), ... respectively since they are single integers and $F^{-1}(0,4) = g(0,4) = 12$ since its of the form (i,n), so we apply g to it..?

4. Bump!