$($Disjoint TMs$)$ Consider the following computational problem: Given TMs $M_1$ and $M_2,$ is it the

case that $L(M_1)L(M_2)=\frac{O}{?}?$ This problem is captured by the language $DIS{J}_{TM}=\{\begin{array}{cc}(\text{:}{M}_1,& M_2\text{:})|\end{array}$

$M_1,M_2$ are TMs and $.$

$($a$)$ If $f$ is a mapping reduction from $A_{TM}$ to $\frac{?}{\text{b}\text{a}\text{r}}(DIS{J}_{TM}),$ what should the inputs and outputs to $f$

be$?($As in Lecture $17,$ slide $18.)$

$($b$)$ Describe a TM computing a mapping reduction from $A_{TM}$ to $\frac{?}{\text{b}\text{a}\text{r}}(DIS{J}_{TM})$ and explain why this

TM is correct.

$($c$)$ Explain why part $($b$)$ implies that $DIS{J}_{TM}$ is not Turing$-$recognizable.

$($d$)$ Is there is a mapping reduction from $A_{TM}$ to $DIS{J}_{TM}?$ Prove your answer.