# Join of recursively enumerable set and its complement

The union of a recursively enumerable set and its complement is $$\Sigma^*$$, which is definitely recursively enumerable. What if instead we consider the following operation, on an RE set $$S$$? $$\{ \# w : w \in S \} \cup \{ \\\ w : w \in \overline{S} \}$$ Is the result recursively enumerable?

• The question in the body doesn't really match the title Apr 9 at 5:47

The operation you described in actually the join operator in the semilattice of Turing degrees, defined as $$A\sqcup B=\{0x | x\in A\}\cup \{1x | x\in B\}$$. It is not hard to show that $$A,B\le_T A\sqcup B$$, thus if $$A\in RE\setminus R$$ then $$\overline{A}\notin RE$$ and thus $$A\sqcup\overline{A}\notin RE$$.