Question

a) List all the languages whose Kleene-star is finite. b) Provide examples of two infinite languages...

a) List all the languages whose Kleene-star is finite.
b) Provide examples of two infinite languages X and Y , and a finite language Z such that X ∩ Y is finite, and XZ is also finite. Explain your answer.

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Answer #1

(a) The empty language ∅ and the null language {ε} which contains blank string are the only two languages with finite Kleene-star or Kleene closure.
(b) Let X = {a, aa, aaa, aaaa, ....} = a*, Y = {a, b, bb, bbb, bbbb, bbbbb, ....} = (a+b)* and Z is an empty language ∅.
Then, although X and Y are infinite languages, there intersection {a} is finite as there is a finite number of strings common to X and Y.
X is an infinite language but Z is a finite language. However, XZ is the empty string ∅ which is also finite. This is because concatenation of all strings of X with all strings of Z provides no string as there is no string in Z.

Hope this helps.

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