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Zorn's lemma can be used to show that every connected graph has a spanning tree. The set of all sub-graphs that are trees is ordered by inclusion, and the union of a chain is an upper bound. Zorn's lemma says that a maximal tree must exist, which is a spanning tree since the graph is connected. [1]
In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset, where "maximal" is with respect to set inclusion.
Zorn's lemma, one of many equivalent statements to the axiom of choice, requires that a partial order in which all chains are upper bounded have a maximal element; in the partial order on the trees of the graph, this maximal element must be a spanning tree. Therefore, if Zorn's lemma is assumed, every infinite connected graph has a spanning tree.
The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem. Set theory. Tarski's theorem about choice: For every infinite set A, there is a bijective map between the sets A and A×A.
Kuratowski proved the Kuratowski-Zorn lemma (often called just Zorn's lemma) in 1922. [6] This result has important connections to many basic theorems. Zorn gave its application in 1935. [7] Kuratowski implemented many concepts in set theory and topology. In many cases, Kuratowski established new terminologies and symbolisms.
The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents). [1] [2] Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. [3]
Zorn's lemma is commonly used with X being a set of subsets; in this case, the upper bound is obtained by proving that the union of the elements of a chain in X is in X. This is the way that is generally used to prove that a vector space has Hamel bases and that a ring has maximal ideals.
A third version of the axiom, also equivalent, is Zorn's lemma. Since the existence of a choice function when X {\displaystyle X} is a finite set is easily proved from axioms 1–8 , AC only matters for certain infinite sets .
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