# hw1 - any other ﬁnite language Problem 3 How many...

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ECS 120: Introduction to the Theory of Computation Homework 1 Due Apr 8, by 1pm in the homework box in Kemper 2131 Problem 1. Let A, B, C be three sets. Prove the following: (a) A \ B = A \ ( B A ). (b) B A if and only if A B = . (c) ( A \ B ) \ C = ( A \ C ) \ ( B \ C ) = A \ ( B C ). (d) A B = and A B = if and only if A = B . Problem 2. For each of the following, give an example language L to prove existence, or explain why no such example exists. If not speciﬁed, assume an underlying alphabet of { 0 , 1 } . Part A. An inﬁnite language with an inﬁnite complement. Part B. An inﬁnite unary language L such that if x L and y L then there is no string in L of length | x | + | y | . Part C. A ﬁnite language having a longest string x that is longer than a longest string of
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Unformatted text preview: any other ﬁnite language. Problem 3. How many diﬀerent languages over the alphabet { a } are accepted by two state DFAs? Name them all. Problem 4. Give DFAs that recognize the following languages. Assume Σ = { , 1 } . (a) The set of all strings with 010 as a substring. (b) The set of all strings which do not have 010 as a substring. (c) The set of all strings which have an even number of 0’s or an even number of 1’s. (d) The complement of { 11 , 111 } . (e) The binary encoding of numbers divisible by 5: { ε, , 00 , 000 , . . . } ◦ { , 101 , 1010 , 1111 , 10100 , . . . } . 1...
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