User:Boris Tsirelson/Sandbox1: Difference between revisions
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In the classical Cantor–Bernstein–Schroeder theorem | In the classical Cantor–Bernstein–Schroeder theorem | ||
* ''X'' and ''Y'' are sets (maybe infinite), | * ''X'' and ''Y'' are sets (maybe infinite), | ||
* "a part" is interpreted as | * "a part" is interpreted as a subset, | ||
* "similar" is interpreted as | * "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]]. | ||
==Notes and references== | ==Notes and references== | ||
{{reflist}} | {{reflist}} | ||
Revision as of 10:13, 1 September 2010
The general idea of the Cantor–Bernstein–Schroeder theorem and related results may be formulated as follows. If X is similar to a part of Y and at the same time Y is similar to a part of X then X and Y are similar. In order to be specific one should decide
- what kind of mathematical objects are X and Y,
- what is meant by "a part",
- what is meant by "similar".
In the classical Cantor–Bernstein–Schroeder theorem
- X and Y are sets (maybe infinite),
- "a part" is interpreted as a subset,
- "similar" is interpreted as equinumerous.