$($a$)$ What do you mean by a decision problem? State the optimization and decision version of the Travelling

Salesperson Problem $($TSP$).$

$($b$)$ Given a set $E$ of problems, define the set E$-$complete. Also, give the steps through which a problem can be

proven E$-$complete.

$($c$)$ The problem of finding a Hamiltonian Cycle is a known NP$-$Complete problem. Prove by reduction that

Hamiltonian Path is also NP$-$Complete. Clearly explain your reduction with the help of an example.