Suppose that $S$ is an infinite countable set, then prove that the power$-$set $2^S$ of $S$ is uncountable. $($hint: use diagonalization$)(5$ pts$)$

Prove that the set of strings $S=\{a,b,c{\}}^{+}$is countable.$($hint: use diagonalization$)(5$ pts$)$

Prove that the halting problem is unsolvable $($undecidable$).($hint: use diagonalization$)(5$ pts$)$

State$-$entry problem: Given a turing machine, an input string and a state in that turing machine, tell whether the turing machine will enter to that state or not during the executing of the input string. Prove that the state$-$entry problem is unsolvable $($undecidable$).($hint: use reduction$)(5$ pts$)$