$5^{th}$ Question:

Decidability and Enumerators. Show that a language Lsube${}^{**}$ is decidable

if and only if there exists an enumerator that enumerates $L$ in shortlex string order $($where strings are arranged by length and within the same length alphabetically$).$ Note: this is an iff statement, you are thus expected to prove both directions.

$6^{th}$ Question:

Large parts of recognizable languages are decidable. Show that every infinite Turing$-$recognizable language has an infinite decidable subset.

Hint: Use problem $5.$

$7^{th}$ Question:

Tint. Define a Turing machine with input alphabet $\{0,1\}$ that decides

$\{1^m{01}^n|m>0$ and $n=2^m\}$