Saturday, June 17, 2017

real analysis - Problem about bijective function



which of the following statement is True ?




1. There exist a bijective function from RQ to R



2 There exist a bijective function from Q to Z×N



3 There does exists strictly increasing function from Z to N



4.There does exists strictly increasing onto function from Z to N



I thinks only option 2 is true because countable map to countable



Answer



1) is true since RQ has same cardinality as R



2) is true.



3) and 4) are false. Suppose there is a strictly incerasing function f:ZN. Let f(0)=n. Then f(1),f(2),.... satisfy the inequalities $... f(-n). But there are only a finite number of integers less than n in N. Hence such an f cannot exist.


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