Regarding cardinality and mapping between sets.

  • #1
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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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  • #2
member 587159
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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  • #3
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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!
 
  • #4
member 587159
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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