If there is an injective function from set A into set B and an injective function from set B into set A, then there is a bijection between A and B. Named for Georg Cantor, Felix Bernstein and Ernst Schroder, it is a fundamental tool for comparing the sizes of infinite sets.
Facts
StatementIf there is an injective function from set A into set B and an injective function from set B into set A, then there exists a bijection between A and B, so the two sets have the same cardinality. 2 Proof YearCantor stated the theorem without proof in 1887; Dedekind proved it independently the same year but never published his proof; the earliest proof publicly presented was Bernstein's, given in Cantor's seminar in 1897 (published 1898); Schroder announced an independent proof in 1896, published 1898, later shown to be flawed. Classification
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Sources
1. Wikipedia: Schröder-Bernstein theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In set theory, the Schröder-Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B.
2. Cantor-Schroder-Bernstein theorem (Wikipedia)
Wikimedia FoundationIntroduction
the Schröder-Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B
History section, 1897 entry
Bernstein, a 19-year-old student in Cantor's Seminar, presents his proof.
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