Assume statements S1 and S2 defined as : S1 : L2-L1 is recursive enumerable where L1 and L2 are recursive and recursive enumerable respectively. S2 : The set of all Turing machines is countable. Which of the following is true ?a)S1 is correct and S2 is not correct.b)Both S1 and S2 are correct.c)Both S1 and S2 are not correct.d)S1 is not correct and S2 is correct.Correct answer is option 'B'. Can you explain this answer?

Question

Answer

S1 : L2-L1 is recursively enumerable where L1 and L2 are recursive and recursively enumerable respectively.

To understand statement S1, let's first define what it means for a language to be recursive and recursively enumerable.

A language L is said to be recursive if there exists a Turing machine that can decide whether a given input belongs to L or not. In other words, the Turing machine will halt and output either 'yes' or 'no' for any given input.

On the other hand, a language L is said to be recursively enumerable (also known as computably enumerable) if there exists a Turing machine that can enumerate (list) all the strings in L. The Turing machine may halt and output a string from L, or it may loop indefinitely without outputting anything for strings not in L.

Now, let's consider the statement S1: L2-L1 is recursively enumerable.

The set difference operation L2-L1 between two languages L1 and L2 is defined as the set of strings that are in L2 but not in L1. In this case, since L1 is recursive and L2 is recursively enumerable, we can conclude that L2-L1 is recursively enumerable. This is because we can design a Turing machine that first enumerates all the strings in L2, and for each enumerated string, checks if it is in L1. If it is not in L1, the Turing machine can output it as part of L2-L1.

Therefore, statement S1 is correct.

S2 : The set of all Turing machines is countable.

To understand statement S2, let's define what it means for a set to be countable.

A set is said to be countable if its elements can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). In other words, there exists a function that can map each element of the set to a unique natural number, and vice versa.

Now, let's consider the set of all Turing machines. We can assign a unique identifier (such as a binary representation) to each Turing machine. Since the set of all Turing machines is countable, we can establish a one-to-one correspondence between each Turing machine and a natural number.

Therefore, statement S2 is correct.

Conclusion:

Both statement S1 and statement S2 are correct.

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