A Turing printer (TP) is a Turing machine with a work tape, a special "print" tape, and a special "print" state. The TM starts with an empty work tape. Every time it enters the "print" state, we consider the current contents of the print-tape to be "printed." If P is a TP, then we write L(P) to denote the set of strings that P eventually prints. (a) Prove that a language L is Turing-recognizable if and only if L = L(P) for some TP P. (b) Prove that a language L is Turing-decidable if and only if L = L(P) for some TP P that prints strings in lexicographic order. (c) Prove that every infinite Turing-recognizable language L contains an infinite subset that is Turing-decidable.