Saturday, March 31, 2018

elementary set theory - Suppose that A1subseteqA and B1subseteqB, and that AsimB and A1simB1. Then AsetminusA1simBsetminusB1.



For A,B,A1,B1 are infinite sets. Suppose that A1A and B1B, and that AB and A1B1. Then AA1BB1.




My attempt:



We denote A and B are equinumerous by AB.


Since AB, there is a bijection f:AB. Similarly, there is a bijection h:A1B1.



From here, I don't know how to proceed to define a bijection from AA1 to BB1. I think the proof may require Axiom of Choice, but don't know how. Please shed some lights!



Answer



A counterexample. Let N={1,2,3,4,} be the natural numbers, and let E={2,4,6,} be the even natural numbers. Then our counterexample is: A1=N,A=N,B1=E,B=N. We have A1B1 and AB but AA1 is empty and BB1 is the set of all odd numbers.


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