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quiz1-solutions-Spring-2007

# quiz1-solutions-Spring-2007 - CS 360 Introduction to the...

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Unformatted text preview: CS 360: Introduction to the Theory of Computing John Watrous, University of Waterloo Solutions to Quiz 1 1. [10 points] Consider the following language: A = { w ∈ { 0, 1 } ∗ : w represents a number in binary notation that is divisible by 3 } . For example 100111 represents the number 39 in binary, so because 39 is divisible by 3 we have 100111 ∈ A . On the other hand, 10011 represents the number 19 in binary, which is not divisible by 3, and so 10011 negationslash∈ A . The string 0 represents the number 0 in binary, so 0 ∈ A . However, any other string starting with 0, such as 011, should be assumed to not be contained in A , because it is not a proper binary representation of a number. Along similar lines, ε negationslash∈ A . Give a DFA that recognizes A . Solution. Here is a DFA that recognizes A : q q 1 q 2 r r 1 r 2 1 0, 1 0, 1 1 1 1 You were not required to explain why or how your DFA works. For an explanation of why the DFA above works, see the homework 1 problem 7 solutions for a similar (but more complicated) version of the problem. Common mistakes. Most people had a good idea about how to do this one, given the similarity to problem 7 on the homework. Some people made small mistakes in calculating remainders, possibly due to time pressure....
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quiz1-solutions-Spring-2007 - CS 360 Introduction to the...

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