Regarding cardinality and mapping between sets.

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Discussion Overview

The discussion revolves around the concept of cardinality and the conditions under which an injection exists between two sets, A and B. Participants explore the implications of the definition of cardinality and the relationship between the existence of injections and cardinality comparisons.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions why it is not always true that if ##\vert A\vert\leq\vert B\vert## then there exists an injection from ##A## to ##B##.
  • Another participant asserts that by definition, ##|A| \leq |B|## implies the existence of an injection from ##A## to ##B##.
  • A participant reflects on their earlier statement about the existence of injections, suggesting that it may have been too strong and expresses difficulty in finding sources that clarify the necessity of the "if and only if" condition.
  • Another participant agrees that the use of "if" in definitions can often lead to confusion, noting that it is common in mathematics to encounter such ambiguities.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of cardinality and injections, indicating that the discussion remains unresolved regarding the necessity of the "if and only if" condition in the definition of cardinality.

Contextual Notes

There are limitations in the discussion regarding the clarity of definitions and the implications of the existence of injections, as well as the potential for confusion arising from the language used in mathematical definitions.

Terrell
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why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?
 
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Terrell said:
why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?

Who told you that?

By definition ##|A| \leq |B|## iff there exists an injection ##A \to B##
 
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Math_QED said:
Who told you that?

By definition ##|A| \leq |B|## iff there exists an injection ##A \to B##
I think saying that me saying that it's not always true is too strong of a statement, but what really happened is I couldn't find any sources that mentions that it must be an if and only if statement. Thank you a lot! It has cause me a lot of unnecessary thinking lol!
 
Terrell said:
I think saying that me saying that it's not always true is too strong of a statement, but what really happened is I couldn't find any sources that mentions that it must be an if and only if statement. Thank you a lot! It has cause me a lot of unnecessary thinking lol!

That's common in mathematics. When one writes a definition one often says "if" while it should be "iff". I remember it confused me in the beginning too!
 
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