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Let ?= (a, b). The Language L = {w E ?. : na(w) < na(w)) is not regular. (Note: na(w) and nu(w) are the number of as and s in tw, respectively.) To show this language is not regular, suppose you are given p. You now have complete choice of w. So choose wa+1, Of course you see how this satisfies the requirements of words in the language. Now, answer the following: (a) What is the largest value of lryl? (Hint: use the pumping lemma) (b) What must you choose for the substring y in the decomposition ayz? (c) Pump up y using i2 and what is the resulting string? (d) Is this string in L and is the language regular?
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