I have been wondering if the following statement is true, $$ A\equiv_TB\iff A'\equiv_TB' $$ where $A, B\subseteq\omega$ and $A'$ denotes the Turing jump of $A$. I have been able to show the forward direction, but have been unable to show the converse. I am starting to think that the converse fails and have been looking for an example. Any suggestions?
Example of sets $A, B$ such that $A', B'$ are Turing equivalent but $A, B$ are not.
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computability