if we define $A_{TM}$ and $\overline{ A_{TM}}$ as follows: $$ \begin{align} A_{TM}&= \{ \langle M,w \rangle |M \text{ is a turing machine and $M$ accepts $w$} \}\\ \overline {A_{TM}}&= \{ \langle M,w \rangle |M \text{ is a turing machine and $M$ do not accepts $w$} \}\end{align} $$ $\overline{A_{TM}}$ and $\Sigma^*\backslash A_{TM}$ both are not turing-recognizeable. right?(a bit confused by the way $\overline{ A_{TM}}$ is defined. noramly $\overline A = \Sigma^*\backslash A$).
My reasons :
1.$\overline{A_{TM}}$ is not reconizable because: if it was then we could run two recognizer for $\overline{A_{TM}}$ and $A_{TM}$ in parallel and decides $A_{TM}$.
2. $\Sigma^*\backslash A_{TM}$ is not recongnizeable by fallowing lemma :
lemma: $L$ and $\overline L$ recognizeable iff $L$ is decideable.
in fact both reasones are quite the same but in second one we are considring inputs ($\langle M,w \rangle$) which are not properly formated.