Is there an example for two subsets A and B of the Reals such that cardinality A = card B and card (R-A) does not equal card( R-B )

Question

Or is there no example in which the cardinalities are not equal?

Answer 1

Take $\;A=\Bbb R\;,\;\;B=Irr=\;$ the irrationals. Then $\;|A|=|B|=2^{\aleph_0}\;$, yet $\;\Bbb R\setminus A=\emptyset\;,\;\;\Bbb R\setminus B=\Bbb Q\ldots\;$

Answer 2

HINT: It can be done with sets $A$ and $B$ that both have the same cardinality as $\Bbb R$ itself.