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NCCS RYK Campus Theory of Automata

NCCS RYK Campus Theory of Automata. Assignment 1 Powered by Muhammad Umair. Task.

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NCCS RYK Campus Theory of Automata

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  1. NCCS RYK CampusTheory of Automata Assignment 1 Powered by Muhammad Umair

  2. Task • Q) Prove that there are as many palindromes of length 2n, defined over Σ = {a,b,c}, as there are of length 2n-1, n = 1,2,3… . Determine the number of palindromes of length 2n defined over the same alphabet as well. Theory Of Automata

  3. Solution • To calculate the number of palindromes of length(2n), consider the following diagram, Theory Of Automata

  4. which shows that there are as many palindromes of length 2n as there are the strings of length n i.e. the required number of palindromes are 3n (as there are three letters in the given alphabet, so the number of strings of length n will be 3n ). Theory Of Automata

  5. To calculate the number of palindromes of length (2n-1) with a as the middle letter, consider the following diagram, Theory Of Automata

  6. which shows that there are as many palindromes of length 2n-1, with a as middle letter, as there are the strings of length n-1, i.e. the required number of palindromes are 3n-1. Similarly the number of palindromes of length 2n-1, with b or c as middle letter, will be 3n-1 as well. Hence the total number of palindromes of length 2n-1 will be 3n-1 + 3n-1 + 3n-1 = 3 (3n-1)= 3n . Theory Of Automata

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