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Prove by induction on n that if L is a language and R is a regular...

Prove by induction on n that if L is a language and R is a regular expression such that L = L(R) then there exists a regular expression Rn such that L(Rn) = L n. Be sure to use the fact that if R1 and R2 are regular expressions then L(R1R2) = L(R1) · L(R2).

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Answer #1

Solution:

Base step: When we have because this ensures . Thus, when , there is a regular expression such that .

Induction hypothesis: Suppose that for an integer there is a regular expression such that .

Induction step: Now consider ; we have

Since product of two regular expressions is a regular expression, letting , we conclude that there is a regular expression with .

By induction we have now proved that for every there is a regular expression such that .

Hope I answered the question.

If you have any doubts, or queries, feel free to ask

I'll respond to you as soon as I can.

Have a nice day

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