Konig's Theorem in set theory is a result of cardinal arithmetic stating that, assuming the axiom of choice, if the cardinality of each set in an indexed family is strictly less than the cardinality of a corresponding set in a second indexed family over the same index set, then the sum of the cardinalities in the first family is strictly less than the product of the cardinalities in the second. It is a foundational inequality of infinite cardinal arithmetic, named for Denes Konig, distinct from his own well-known theorem on bipartite graphs.
Facts
StatementIf the axiom of choice holds and, for every index i in a set I, the cardinal number kappa-i is strictly less than the cardinal number lambda-i, then the sum of the kappa-i over I is strictly less than the product of the lambda-i over I. 1 Classification
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Source Kőnig's theorem (set theory) (Wikipedia)
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1. Konig's theorem (set theory), Wikipedia
Introduction
Konig's theorem states that if the axiom of choice holds, I is a set, ki and lambda-i are cardinal numbers with ki less than lambda-i, for every i in I, then the sum of the ki is strictly less than the product of the lambda-i.
Lead section
Kőnig's theorem was introduced by Kőnig (1904) in the slightly weaker form that the sum of a strictly increasing sequence of nonzero cardinal numbers is less than their product.
View the SourceKőnig's theorem (set theory) (Wikipedia)
In Branch: Set Theory, Lead
In set theory, Kőnig's theorem states that if the axiom of choice holds
In Group: Set Theory, Lead section
In set theory, Kőnig's theorem states that if the axiom of
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