# Proof of Axiom of Choice equivalent.

• leetaxx0r
In summary, the conversation revolves around proving the equivalence between the axiom of choice and a statement involving functions and sets. The speaker was able to prove one direction but is struggling with the other. There is a discussion about equivalence relations and how they can be used to prove the statement. The conversation ends with a possible solution involving pairwise disjoint sets.
leetaxx0r
I'm trying to prove that the axiom of choice is equivalent to the following statement:

For any set $$X$$ and any function $$f:X\to X$$, there exists a function $$g:X\to X$$ such that $$f\circ g\circ f=f$$.

I was able to prove that the AoC implies this, but I'm having a harder time going the other direction. It seems like if you define an equivalence relation on $$X$$ where $$x\sim y$$ iff $$f(x)=f(y)$$, then the composite function $$g\circ f$$ must map everything in each equivalence class to one of its members.

This seems like it's important, but we're still only choosing points for a very specific collection of subsets of $$X$$, namely the equivalence classes induced by the function. Is there a way to extend this to any collection of subsets, or am I heading in the wrong direction?

Hmm... this is kind of a weird way of doing it, but I think it works: Let A be a pairwise disjoint family of nonempty sets. Then Let X be (UA)U(A)U{null}. Then define f:X->X by letting f(x)=B if x is in UA and B is in A and x is in B, and letting f(B)=null for B in A, and letting f(null)=null.

Then the equivalence relations include elements of A, since for every element B of A, everything in B and nothing else maps to B.

That looks very clean. Thank you very much. It was the fact that the domain and codomain had to be equal that was causing most of the trouble it seems.

## 1. What is the Axiom of Choice?

The Axiom of Choice is a fundamental mathematical principle that states that given any collection of non-empty sets, it is possible to choose one element from each set. It is used in various fields of mathematics, including set theory, topology, and functional analysis.

## 2. Why is the Axiom of Choice important?

The Axiom of Choice allows for the creation of new mathematical objects, such as infinite sets and functions, and helps to prove many important theorems in mathematics. It also has applications in computer science, economics, and other fields.

## 3. What is meant by "Proof of Axiom of Choice equivalent"?

A proof of Axiom of Choice equivalent means that the Axiom of Choice is equivalent to another mathematical statement or principle. This means that both statements have the same logical strength and can be used interchangeably in mathematical proofs.

## 4. What are some examples of statements that are equivalent to the Axiom of Choice?

The Well-Ordering Theorem, the Zorn's Lemma, and the Hahn-Banach Theorem are some examples of statements that are equivalent to the Axiom of Choice. These statements have been shown to have the same logical strength as the Axiom of Choice and can be used in place of it in mathematical proofs.

## 5. Why is the Axiom of Choice a controversial topic in mathematics?

The Axiom of Choice has been a topic of much debate and controversy among mathematicians. Some argue that it is a necessary and intuitive principle that allows for the development of mathematics, while others argue that it leads to counterintuitive results and undermines the foundations of mathematics. The controversy surrounding the Axiom of Choice has led to the development of alternative set theories, such as constructive and finitistic mathematics.

• Set Theory, Logic, Probability, Statistics
Replies
5
Views
687
• Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
9
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
3
Views
880
• Calculus
Replies
9
Views
586
• Set Theory, Logic, Probability, Statistics
Replies
10
Views
4K
• Set Theory, Logic, Probability, Statistics
Replies
9
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
21
Views
11K
• Set Theory, Logic, Probability, Statistics
Replies
10
Views
2K