Final answer:
The class of TM-decidable languages is closed under union, concatenation, star, intersection, and complement. The closure under each operation can be shown by constructing corresponding Turing machines.
Step-by-step explanation:
The class of TM-decidable languages refers to the set of languages that can be decided by a Turing machine. We can show that this class is closed under various operations:
a) Union:
Let L1 and L2 be TM-decidable languages.
To show closure under union, we construct a Turing machine that simulates both machines for each input, accepting if either machine accepts.
Hence, the union of two TM-decidable languages is also TM-decidable.
b) Concatenation:
Let L1 and L2 be TM-decidable languages.
To show closure under concatenation, we construct a Turing machine that simulates both machines on all possible divisions of the input, accepting if both machines accept.
Hence, the concatenation of two TM-decidable languages is also TM-decidable.
c) Star:
Let L be a TM-decidable language.
To show closure under star, we construct a Turing machine that simulates the machine multiple times, accepting if the input can be divided into parts that are accepted by each simulation.
Hence, the star of a TM-decidable language is also TM-decidable.
d) Intersection:
Let L1 and L2 be TM-decidable languages.
To show closure under intersection, we construct a Turing machine that simulates both machines for each input, accepting only if both machines accept.
Hence, the intersection of two TM-decidable languages is also TM-decidable.
e) Complement:
Let L be a TM-decidable language.
To show closure under complement, we construct a Turing machine that simulates the machine on each input, accepting if the original machine rejects and vice versa.
Hence, the complement of a TM-decidable language is also TM-decidable.