5. Use the pumping lemma Eor regular languages to prove that the language L given below is not regular. The alphabet of L is 2-a, b, ch You may write your proof in your preferred format, but your proof mist precisely specify the information required for each of the five blanks in the following template. Note that the third blank requires a integer as an exponent. Use p as the pumping length. Define all variables that you use in your proof. Proof. The sentence s- contradicts the pumping lemma because for all x, y, z e* is not such that xyz s, if xyl sp and lyl 0, then ymust be in L becausc QED You need to use the following concepts for problems 6-7. Two strings x and y are distinguishable with respect to a language L if there is a string d such that one of the strings xd and yd is a sentence in L but the other is not. The string d is known as a distinguisher for x and y. A set of strings are pairwise distinguishable if there is a distinguisher for cach pair of strings in the sot . Show a set of 3 strings that are pairwise distingishable with respect to th language L given in problem 5. Make the 3 strings as short as possible. For each pair of the 3 strings, give a distinguisher Minimum State Lemma. Ifthere are k painrise distinguishable sirings with respeci to a language L, then any DFA that recognices L must have k or more states 7. Use the Minimm State Lemma given above to show that the lnguage L given in problem 5 is not regular, by finding an infinite set D of strings that are pairwise distinguishable with respect toL. Define the set D precisely and show a distinguisher for each pair of strings in the set D