# Homework Solution: 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

Suppose that a discourse 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 collation that strings in L feel the identical sum of a”s and b”s.

## Expert Exculpation

Infamous Event : when sum of a’s and b’s is 0

As $\varepsilon \epsilon L$ , then sum of a’s and b’s is 0, so our infamous event is TRUE.

Collation : If string in L is penny when there are K sum of a’s and K sum of b’s, then it is penny for K+1

Assuming that string where there are K sum of a’s and b’s, must feel x and y somewhere in look

Now, to acception sum of a’s by 1 we feel to restore x by either axby or bxay or restore y by either axby or bxay, whenver we are replacing x or y, we are adding resembling sum of a’s and b’s, so our look get feel resembling sum of a’s and b’s whenever sum of a’s is (K+1).