Explain Equivalence relation & Digitalization with suitable Example. Construct a DFA with reduced states equivalent to the regular expressions: 10 + (0 + 11) 0 * 1 Explain Prefix, suffix, proper prefix and proper suffix of a string with suitable Example. Is it possible for two distinct sets A and B to satisfy the equation A X B=B X A? If so, under what circumstances? Give reason for your answer. Is it possible for two distinct sets A and B to satisfy A X B B X A? If so, under what circumstances? Give reason for your answer. Construct a FA, which does not contain ‘101’ and further minimize it. Write a short note on Derivation Trees. Write a short note on Application of Pumping Lemma Write a short note on Regular expression.
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1. Prove the Distributive and De Morgan’s laws for logic. 2. Give a direct proof for the following: Suppose x is an integer. Prove that if x is odd, then x+1 is even. 3. Prove the following, using the proof by cases method: Suppose x is a real number, then -|x| ≤ x ≤ |x|. 4. Prove the following, using the proof by contradiction method. Suppose x and y are odd integers, then xy is odd. 5. What, if anything, is wrong with the following proof: Suppose that m is an integer. Prove that if m 2 is odd, then m is odd. Proof: Assume m is odd. Then m = 2k+1 for some integer k. Therefore m 2 = (2k+1) 2 =4k 2 +4k+1 = 2(2k 2 +2k) +1, which is odd. Therefore if m 2 is odd, then m is odd. 6. Given the premises s→(o∧r) and n∧¬o prove (using the rules of inference) the conclusion that ¬s. Be sure to label your steps and give the rules of inference you are using. 7. Given the premise ∃x(p(x)∧q(x)) prove (using the rules of inference ) the conclusion that ∃x p(x)∧∃x q(x). Be sure to label your steps and give the rules of inference you are using.
