
TL;DR
This paper presents a simple proof that for finite sets, multiple injections imply a single injection, and from this derives that multiple bijections imply a single bijection, clarifying a long-standing mathematical history.
Contribution
It provides a straightforward proof connecting finite multiple injections and bijections, resolving a complex historical debate in set theory.
Findings
nA<=nB implies A<=B for finite sets
nA==nB implies A==B for finite sets
Clarifies the relationship between multiple injections and bijections in finite sets
Abstract
Write A<=B if there is an injection from A to B, and A==B if there is a bijection. We give a simple proof that for finite n, nA<=nB implies A<=B. From the Cantor-Bernstein theorem it then follows that nA==nB implies A==B. These results have a long and tangled history, of which this paper is meant to be the culmination.
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Taxonomy
TopicsHistory and Theory of Mathematics · semigroups and automata theory · Mathematics and Applications
