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1. For each of the following languages, prove that it is not regular. {an :n>0} (b) L= {abi 2j} L2 (a) L = (c)

1. For each of the following languages, prove that it is not regular. {an :n>0} (b) L= {abi 2j} L2 (a) L = (c) L3= {w {a,b}* : na(w) #ni(w)} = 2. Given a language L over the alphabet , define f(L) = {vo v L, 0 , and v=uo for some u [*} Prove that f(L) is regular if L is regular.

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