Recursively enumerable notation $RE$ vs. $RE\setminus R$

Question

I know that it's a bit stupid question.. , but still, Is there any difference between $RE$ and $RE\setminus R$ notations?

I'm asking because I saw that in some places using both of the notations, for example for $$ L_{HP}=\{(\langle M\rangle,x)|M\ halts\ on\ x\}$$

Answer 1

$RE$ stands for recursively enumerable (recognizable) and $R$ for recursive (decidable). $RE\ \backslash\ R$ simply stands for all the sets that are are recursively enumerable, but not recursive (recall that every recursive set and its complement are recursively enumerable, but not every recursively enumerable set is recursive e.g. Halting Problem).

$R \subset RE$ and $L_{HP} \notin R$, therefore $L_{HP}$ can be in both $RE$ and $RE\ \backslash\ R$.

Answer 2

If I understood it right:

$$L_{HP}\in RE\setminus R$$

and because $$RE\setminus R\subset RE$$ So necessarily: $$L_{HP}\in RE$$