$4.$ Examples of languages $(7$ points$)$: For each part of the question, use precise mathematical notation or English to define your examples and then briefly justify why they work. $($a$)($Graded for correctness$)$ Two undecidable languages L$1$ and L$2$ over the same alphabet whose intersection L$1$ L$2$ is decidable, or write NONE if there is no such example $($and explain why$).($b$)($Graded for correctness$)$ A regular language L$3$ and an unrecognizable language L$4$ over the same alphabet whose set$-$wise concatenation L$3$ L$4$ is unrecognizable, or write NONE if there is no such example $($and explain why$).($c$)($Graded for completeness$)$ A co$-$recognizable language L$5$ that is NP$-$complete, or write NONE if there is no such example $($and explain why$).$ Recall the definition: A language L over an alphabet $$ is called co$-$recognizable if its complement, defined as $\backslash$ L $=\{$x in $|$ x $/$ in L$\},$ is Turing$-$recognizable.