Let S be the set 0p p is a prime number Prove that S is not

Let S be the set

{0^p | p is a prime number}.

Prove that S is not regular.

Solution

We know that a set is said to be a regular set if it exbits the values of regular expression.

Let us write a few elements of set S to see if we have a regular expression

{ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,.....)

We can see that there are 4 elements of string length one and 21 elements of string length two. Likewise, there are 143 elements of sting length three. Therefore, the set values do not exibit any regular expression. hence, the given set S is not regular.