Turing reducibility and information

$($a$)$ Let $=\{0,1\}$ be the alphabet.

Statement: For any two languages A and $B,$ such that $A,\frac{B}{?}=\frac{O}{,}{}^{**},$ a

language $C$ exists, where $A{?}_{T}C$ and $B{?}_{T}C.$

Is the statement true?

Circle the answer:

yes

no

Statement: For any two languages A and $B,$ such that $A,\frac{B}{?}=\frac{O}{,}{}^{**},$ a

language $C$ exists, where $C{?}_T$A and $C{?}_{T}B.$

Is the statement true?

Circle the answer:

yes

no

$($b$)$ If TM $M$ generates $x$ on input $w_{{?}_{?}}$ then the pair $M$ and $w$ is considered a description

of $x.$ We can write this description wit Close usual notation for encoding several

objects into a single binary string $($:$M,w$:$).$

Statement: We define strings of the form $($:$M$:$)$ and $($:$M,w$:$)$ such that there exists a

constant $c>$ for which the following inequality holds

$|($:$M,w$:$)||($:$M$:$)|+|w|+c$

for any pair of a TM $M$ and string $w.$

Is the statement true?

Circle the answer:

yes

no

$($c$)$ Consider the following Turing machine $M,$ which expects an input of the form

$($:$N$:$),$ where $N$ is a TM and $w$ is an input for it$.$

$M="$on input $($:$Nw$:$),$ where $N$ is a TM and $w$ is a string:

i$.$ run $N$ on $w$ until it halts and produces an output string $s$

ii$.$ output the string $SS{S}^{\text{'}\text{'}}$