User:Boris Tsirelson/Sandbox1: Difference between revisions
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* what is meant by "a part", | * what is meant by "a part", | ||
* what is meant by "similar". | * what is meant by "similar". | ||
In the classical Cantor–Bernstein–Schroeder theorem | In the classical Cantor–Bernstein–Schroeder theorem | ||
* ''X'' and ''Y'' are [[Set (mathematics)|sets]] (maybe infinite), | * ''X'' and ''Y'' are [[Set (mathematics)|sets]] (maybe infinite), | ||
* "a part" is interpreted as a [[subset]], | * "a part" is interpreted as a [[subset]], | ||
* "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]]. | * "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]]. | ||
Not all statements of this form are true. For example, let | |||
* ''X'' and ''Y'' are [[triangle]]s, | |||
* "a part" means a triangle inside the given triangle, | |||
* "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles"). | |||
Then the statement fails badly: every triangle ''X'' evidently is similar to some triangle inside ''Y'', and the other way round; however, ''X'' and ''Y'' need no be similar. | |||
==Notes and references== | ==Notes and references== | ||
{{reflist}} | {{reflist}} | ||
Revision as of 10:28, 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.
Not all statements of this form are true. For example, let
- X and Y are triangles,
- "a part" means a triangle inside the given triangle,
- "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles").
Then the statement fails badly: every triangle X evidently is similar to some triangle inside Y, and the other way round; however, X and Y need no be similar.