Sunday, 20 April 2014

elementary set theory - Use of Cantor Schroder-Bernstein theorem?


Use the Schröder-Bernstein theorem to show that there is a bijection between two intervals [0,1]R and [1,)R, thus they have the same cardinality.




What about the sets (0,5) and (10,20)? Is there a bijection between them? (Don't need the Schröder-Bernstein theorem here). Similarly, consider [0,) and [1,).




Hello, this is a question from my practice final. Can anyone explain how to answer this question? As I understand, the theorem allows you to find a one-to-one function between two intervals to show that they have same cardinality, but I don't know how to apply this. For example, for the first question, is it simple as a function like f(x)=1/x or is there something more than that?

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