Difference between revisions of "Axiom of choice"

(Created page with "The '''Axiom of choice''' is an axiom of set theory. The axiom of choice says that if one is given any collection of boxes, each containing at least one object, it is...")
 
m
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
The '''Axiom of choice''' is an [[axiom]] of [[set theory]]. The axiom of choice says that if one is given any collection of boxes, each containing at least one object, it is possible to make a selection of exactly one object from each box, even if the collection is infinite.  
 
The '''Axiom of choice''' is an [[axiom]] of [[set theory]]. The axiom of choice says that if one is given any collection of boxes, each containing at least one object, it is possible to make a selection of exactly one object from each box, even if the collection is infinite.  
  
Wikipedia states it as ,"Formally, it states that for every indexed family {\displaystyle (S_{i})_{i\in I}} (S_{i})_{i\in I} of nonempty sets there exists an indexed family {\displaystyle (x_{i})_{i\in I}} (x_{i})_{i\in I} of elements such that {\displaystyle x_{i}\in S_{i}} x_{i}\in S_{i} for every {\displaystyle i\in I} i\in I."
+
Formally, the Axiom of choice says that, given a non-empty set <math>S</math> of non-empty sets, there is a choice function on <math>S</math>. That is, there is a function <math>f:S\rightarrow\bigcup\limits_{A\in S}A</math> such that <math>f(A)\in A</math> for each <math>A\in S</math>. It is also equivalent to the statement that, given a set <math>S</math> of non-empty sets, <math>\prod\limits_{A\in S}A</math> is non-empty. An equivalent form of the Axiom of choice says, given a set <math>S</math> of non-empty pairwise disjoint sets, there exists a set <math>X</math> with one element from each set in <math>S</math>.
 +
 
 +
The Axiom of choice is equivalent to [[Zorn's Lemma]] and the [[Well-Ordering theorem]] assuming [[Zermelo-Fraenkel Axioms]].
  
 
It was discovered by German mathematician, Ernst Zermelo in 1904.
 
It was discovered by German mathematician, Ernst Zermelo in 1904.
 +
 +
{{stub}}
 +
 +
[[Category:Set theory]]

Revision as of 23:10, 1 June 2019

The Axiom of choice is an axiom of set theory. The axiom of choice says that if one is given any collection of boxes, each containing at least one object, it is possible to make a selection of exactly one object from each box, even if the collection is infinite.

Formally, the Axiom of choice says that, given a non-empty set $S$ of non-empty sets, there is a choice function on $S$. That is, there is a function $f:S\rightarrow\bigcup\limits_{A\in S}A$ such that $f(A)\in A$ for each $A\in S$. It is also equivalent to the statement that, given a set $S$ of non-empty sets, $\prod\limits_{A\in S}A$ is non-empty. An equivalent form of the Axiom of choice says, given a set $S$ of non-empty pairwise disjoint sets, there exists a set $X$ with one element from each set in $S$.

The Axiom of choice is equivalent to Zorn's Lemma and the Well-Ordering theorem assuming Zermelo-Fraenkel Axioms.

It was discovered by German mathematician, Ernst Zermelo in 1904.

This article is a stub. Help us out by expanding it.