Prove or disprove: $($a$)$ L $$T L for any language L$.$

$(4$ pts$)($b$)$ Suppose that L$1$ is recognizable and L$2$ is decidable. State, with proof, whether L$1$ L$2$ is

necessarily recognizable, and whether it is necessarily decidable.

$(4$ pts$)($c$)$ Define LLOOP $=\{($M$,$ x$)$ : M is a TM that loops on input x$\}.$ State, with proof,