Intereting Posts

Under what conditions does a ring R have the property that every zero divisor is a nilpotent element?
Show that $\frac{(-1)^n}{n+(-1)^n\sqrt{n+1}}=\frac{(-1)^n}n+\mathcal{O}\left(\frac{1}{n^{3/2}}\right)$
inverse Laplace Transform: $ L^{-1} \{\log \frac{s^2 – a^2}{s^2} \}$.
Evaluate: $\int_{0}^{\pi}\frac{\cos 2017x}{5-4\cos x}dx$
How to generalize symmetry for higher-dimensional arrays?
Asymptotic (divergent) series
If $\sum\limits_{j=1}^n | w_j|^2 \leq 1$ implies $ \left| \sum\limits_{j=1}^n z_j w_j \right| \leq 1$, then $\sum\limits_{j=1}^n | z_j|^2 \leq 1$
What's the asymptotic lower bound of the sum $\frac 3 2 + \sum_{k=3}^{n} \frac{n!}{k(n-k)!n^k}$?
Quadratic Variation and Semimartingales
Albert, Bernard and Cheryl popular question (Please comment on my theory)
Tensor product of Hilbert Spaces
Mirror anamorphosis for Escher's Circle Limit engravings?
What is the pattern in these ordering properties?
If $n$ is squarefree, $k\ge 2$, then $\exists f\in\Bbb Z : f(\overline x)\equiv 0\pmod n\iff \overline x\equiv \overline 0\pmod n$
Archimedean places of a number field

Let $A,B$ be two sets. The Cantor-Schroder-Bernstein states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$.

I was wondering whether the following statements are true (maybe by using the AC if necessary):

- Suppose $f \colon A\to B$ and $g\colon B\to A$ are both surjective, does this imply that there is a bijection between $A$ and $B$.
- Suppose either $f\colon A\to B$ or $g\colon A\to B$ is surjective and the other one injective, does this imply that there is a bijection between $A$ and $B$.

- Uncountable subset with uncountable complement, without the Axiom of Choice
- (ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior
- Axiom of Choice, Continuity and Intermediate Value Theorem
- Why uniform closure $\mathscr{B}$ of an algebra $\mathscr{A}$ of bounded complex functions is uniformly closed?
- Is there a known well ordering of the reals?
- Cauchy functional equation with non choice

- ZF and the Existence of Finitely additive measure on $\mathcal{P}(\mathbb{R})$
- Is $\cos(\alpha x + \cos(x))$ periodic?
- Maximum of the Variance Function for Given Set of Bounded Numbers
- Number of well-ordering relations on a well-orderable infinite set $A$?
- injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$
- determine whether $f(x, y) = \frac{xy^3}{x^2 + y^4}$ is differentiable at $(0, 0)$.
- The cardinality of a countable union of countable sets, without the axiom of choice
- Drawing by lifting pencil from paper can still beget continuous function.
- If $g \circ f$ is surjective, show that $f$ does not have to be surjective?
- The equivalence of “Every surjection has a right inverse” and the Axiom of Choice

For the first one the need of the axiom of choice is essential. There are models of ZF such that $A,B$ are sets for which exists surjections from $A$ onto $B$ and vice versa, however there is no bijection between the sets.

Using the axiom of choice we can simply inverse the two surjections and have injections from $A$ into $B$ and vice versa, then we can use Cantor-Bernstein to ensure a bijection exists.

The second one, I suppose should be $f\colon A\to B$ injective and $g\colon A\to B$ surjective, again we need the axiom of choice to ensure that there is a bijection, indeed there are several models without it where such sets exist but there is no bijection between them. Using the axiom of choice we reverse the surjection and use Cantor-Bernstein again.

It should be noted that without the axiom of choice it is true that if $f\colon A\to B$ is injective then there is $g\colon B\to A$ surjective. Therefore if the first statement is true, so is the second, and if the second is false then so is the first.

Another interesting point on this topic is this: The **Partition Principle** says that if there is $f\colon A\to B$ surjective then there exists an injective $g\colon B\to A$. Note that we do not require that $f\circ g=\mathrm{id}_B$, but simply that such injection exists.

This principle implies both the statements, and is clearly implied by the axiom of choice. It is open for over a century now whether or not this principle is equivalent to the axiom of choice or not.

Lastly, as stated $f\colon A\to B$ injective and $g\colon B\to A$ surjective cannot guarantee a bijection between $A$ and $B$ with or without the axiom of choice. Indeed the identity map is injective from $\mathbb Z$ into $\mathbb R$, as well the floor function, $x\mapsto\lfloor x\rfloor$ is surjective from $\mathbb R$ to $\mathbb Z$ but there is no bijection between $\mathbb Z$ and $\mathbb R$.

As stated, 2. is clearly false (just take $A=\{ 0,1\} ,B=\{ 0\}$ with $f$ identically zero, and $g$ likewise). I will assume that it’s actually $f:A\to B$ and $g:A\to B$.

Using axiom of choice, both statements can be shown to be true, simply because when we have a surjection $f:A\to B$, then by axiom of choice we can choose a right inverse $f^{-1}:B\to A$ which will be injective, so we can reduce both statements to the usual C-B-S.

Without choice, neither statement can be proved.

For the first one, see https://mathoverflow.net/questions/38771 (apparently, it would imply countable choice).

For the second one, see https://mathoverflow.net/questions/65369/half-cantor-bernstein-without-choice.

- New Idea to prove $1+2x+3x^2+\cdots=(1-x)^{-2}$
- Can an odd perfect number be a nontrivial multiple of a triangular number?
- Proofs for an equality
- why is the following thing a projection operator?
- Can one construct a “Cayley diagram” that lacks only an inverse?
- For any rng $R$, can we attach a unity?
- Trigonometric Uncertainty Propagation
- Sequence in $C$ with no convergent subsequence
- induced representation, dihedral group
- If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
- Finding an $f(x)$ that satisfies $f(f(x)) = 4 – 3x$
- Irreducible elements in $\mathbb{Z}$ and is it a Euclidean domain?
- Composition of functions question
- Uniform convergence of infinite series
- An asymptotic term for a finite sum involving Stirling numbers