# Answered Essay: Suppose that a language L Subsetequalto {a, b}*is defined as follows: (1) epsilon element

Suppose that a language L Subsetequalto {a, b}*is defined as follows: (1) epsilon elementof L, (2) If x, y elementof L, then so are axby and bxay. (3) Nothing else is in L. Prove by induction that strings in L have the same number of a”s and b”s.

Base Case : when number of a’s and b’s is 0

As $\varepsilon \epsilon L$ , then number of a’s and b’s is 0, so our base case is TRUE.

Induction : If string in L is true when there are K number of a’s and K number of b’s, then it is true for K+1

Assuming that string where there are K number of a’s and b’s, must have x and y somewhere in expression

Now, to increase number of a’s by 1 we have to replace x by either axby or bxay or replace y by either axby or bxay, whenver we are replacing x or y, we are adding equal number of a’s and b’s, so our expression will have equal number of a’s and b’s whenever number of a’s is (K+1).

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