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Introduction to the Theory of Computation

Sipser Michael

Chapter 0: Chapter 0

Problem 1

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Examine the following formal descriptions of sets so that you understand which members they contain. Write a short informal English description of each set. a. $\\{1,3,5,7, \ldots\\}$ b. $\\{\ldots,-4,-2,0,2,4, \ldots\\}$ c. $\\{n \mid n=2 m$ for some $m$ in $\mathcal{N}\\}$ d. $\\{n \mid n=2 m$ for some $m$ in $\mathcal{N}$, and $n=3 k$ for some $k$ in $\mathcal{N}\\}$ e. $\\{w \mid w$ is a string of os and 1 s and $w$ equals the reverse of $w\\}$ f. $\\{n \mid n$ is an integer and $n=n+1\\}$

Short Answer

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a. Odd positive integers b. All even integers (positive, negative, and 0) c. Even positive integers d. Multiples of 6 e. Binary palindromes f. Empty set (∅)

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: a. Set A: {1, 3, 5, 7, ...}

This set contains all odd positive integers. They can be described as integers which are not divisible by 2.

Step 2: b. Set B: {..., -4, -2, 0, 2, 4, ...}

This set contains all even integers, both positive and negative, as well as 0. They can be described as integers which are divisible by 2.

Step 3: c. Set C: {n | n = 2m for some m in ℕ}

This set contains all the even positive integers. It contains the numbers obtained by multiplying any natural number m by 2.

Step 4: d. Set D: {n | n = 2m for some m in ℕ, and n = 3k for some k in ℕ}

This set contains all positive integers that are both multiples of 2 and 3. These numbers can also be described as multiples of 6, since the least common multiple of 2 and 3 is 6.

Step 5: e. Set E: {w | w is a string of 0s and 1s and w equals the reverse of w}

This set contains binary strings (strings made up of 0s and 1s) which read the same forwards and backwards. These strings are called palindromes.

Step 6: f. Set F: {n | n is an integer and n = n + 1}

This set is actually empty (denoted by ∅), as it contains no elements. The given condition (n = n + 1) is not possible for any integer, as it would mean that 1 is equal to 0, which is a contradiction.

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Chapter 0: Chapter 0

Short Answer

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Step 1: a. Set A: {1, 3, 5, 7, ...}

Step 2: b. Set B: {..., -4, -2, 0, 2, 4, ...}

Step 3: c. Set C: {n | n = 2m for some m in ℕ}

Step 4: d. Set D: {n | n = 2m for some m in ℕ, and n = 3k for some k in ℕ}

Step 5: e. Set E: {w | w is a string of 0s and 1s and w equals the reverse of w}

Step 6: f. Set F: {n | n is an integer and n = n + 1}

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

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