Tuesday, 30 August 2016

elementary set theory - Cardinality of mathbbR and mathbbR2

I am working on this exercise for an introductory Real Analysis course:




Show that |R| = |R2|.




I know that R is uncountable. I also know that two sets A and B have the same cardinality if there is a bijection from A onto B. So if I show that there exists a bijection from R onto R2 then I beleive that shows that |R| = |R2|.



Let xiR, where each xi is expressed as an infinite decimal, written as xi=xi0.xi1xi2xi3...,. Each xi0 is an integer, and xik{0,1,2,3,4,5,6,7,8,9}. Then, let




f(xi)=(xi0.xi1xi3xi5...,xi0.xi2xi4xi6...)



What should I do to show that f:RR2 is an injective function? Any suggestions or help with the question would be appreciated.

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