Question

a) What is the Kleene closure of a set of strings?

b) Find the Kleene closure of the set\(\left\{ {{\bf{11,}}\,\,{\bf{0}}} \right\}\).

Short answer

a) Therefore, the definition of Kleene closure is “The set consisting of a sequence of arbitrarily many strings from A. That is,\({\bf{A* = }}\bigcup\limits_{{\bf{k = 0}}}^\infty {{{\bf{A}}^{\bf{k}}}} \)

b) Hence, the set \(\left\{ {{\bf{11,}}\,\,{\bf{0}}} \right\}\) is a Kleene closure of the set.

Step-by-step solution

General form

Definition of Kleene closure: Suppose that A is a subset of\({\bf{V*}}\). Then the Kleene closure of A, denoted by\({\bf{A*}}\), is the set consisting of sequences of arbitrarily many strings from A. That is,\({\bf{A* = }}\bigcup\limits_{{\bf{k = 0}}}^\infty {{{\bf{A}}^{\bf{k}}}} \).

Step 2: Describe the Kleene closure

Kleene closure:

The set consists of sequences of arbitrarily many strings from A. That is,\({\bf{A* = }}\bigcup\limits_{{\bf{k = 0}}}^\infty {{{\bf{A}}^{\bf{k}}}} \)

Hence, the definition of Kleene closure is shown above.

Finding the Kleene closure of the set

The given set is\(\left\{ {{\bf{11,}}\,\,{\bf{0}}} \right\}\).

Then, any number of zeros may be used in the Kleene closure of the set \(\left\{ {{\bf{11,}}\,\,{\bf{0}}} \right\}\)but an even number of ones must also be included (as the 1s occur in pairs).

The Kleene closures contain all bit strings with an even number of ones where the ones appear in pairs; despite the fact that the order of the zeros and ones does not matter (0s have no restrictions).

So, the given set is a Kleene closure of the set.

Hence, the set \(\left\{ {{\bf{11,}}\,\,{\bf{0}}} \right\}\) is a Kleene closure of the set.