Reductions

We say $A⩽B$ if there exists a map $\sigma :{\Sigma }^{\ast }\to {\Sigma }^{\ast }$ such that $s\in A⟺\sigma (s)\in B$. $\sigma$ is called a reduction of $A$ to $B$. If $A⩽B$,

1. $\overline{A}⩽\overline{B}$,
2. $B\in \text{RE}⟹A\in \text{RE}$, $A\notin \text{RE}⟹B\notin \text{RE}$
3. $B\in \text{co-RE}⟹A\in \text{co-RE}$, $A\notin \text{co-RE}⟹B\notin \text{co-RE}$