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# sample-quiz-1 - 8 Problem 8 A rational number is a real...

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CPS 102: Discrete Mathematics Instructor: Bruce Maggs Quiz 1 Date: Monday October 4, 2010 NAME: Prob #. Score Max Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1

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Problem 1 Use a regular expression to describe the language accepted by the following deterministic finite automata (DFA). 2
Problem 2 Draw a DFA that accepts the language ε + ( aab * a ) + ( bba * ). 3

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Problem 3 Use a regular expression to describe the set of strings over the alphabet { 0 , 1 } in which every 1 is immediately followed by a zero. 4
Problem 4 Draw a DFA that accepts the set of strings of 0’s and 1’s that contain at least one instance of three consecutive 0’s. 5

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Problem 5 Prove that the set { 01 , 01001 , 010010001 , 01001000100001 , . . . } cannot be accepted by any DFA. 6
Problem 6 Show that the number of different languages over the alphabet Σ = { 0 , 1 } that are accepted by deterministic finite automata with only two states is finite. 7

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Problem 7 Prove by contradiction: There are infinitely many even numbers.

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Unformatted text preview: 8 Problem 8 A rational number is a real number that can be expressed as the ratio of two integers. An irrational number is a real number that is not rational. Provide an indirect proof of the following statement, i.e., prove the contrapositive. If a and b are real numbers and a · b is an irrational number, then either a or b is irrational. 9 Problem 9 Prove by induction that 1 1 · 3 + 1 3 · 5 + 1 5 · 7 + ··· + 1 (2 n-1)(2 n + 1) = n 2 n + 1 . 10 Problem 10 Suppose that there are only two types of postage stamps, 4-cent stamps and 5-cent stamps. Prove that any amount of postage of 12 cents or greater can be made up out of 4-cent and 5-cent stamps. Hint: It is possible to prove this using strong induction over N . 11...
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