For A,B,A1,B1 are infinite sets. Suppose that A1⊆A and B1⊆B, and that A∼B and A1∼B1. Then A∖A1∼B∖B1.
My attempt:
We denote A and B are equinumerous by A∼B.
Since A∼B, there is a bijection f:A→B. Similarly, there is a bijection h:A1→B1.
From here, I don't know how to proceed to define a bijection from A∖A1 to B∖B1. 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 A1∼B1 and A∼B but A∖A1 is empty and B∖B1 is the set of all odd numbers.
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