Mathematics All Around (6th Edition)
6th Edition
ISBN: 9780134434681
Author: Tom Pirnot
Publisher: PEARSON
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Chapter 5.4, Problem 5E
To determine
Whether the statement
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In this exercise, we will investigate a technique to prove that a language is notregular. This tool is called the pumping lemma.The pumping lemma says that if M = (S, I, f, s0, F ) is a DFA with p states (i.e., p = |S|) and if the wordw is in L(M ) (the language generated by M ) and w has length greater than or equal to p, then w may bedivided into three pieces, w = xyz, satisfying the following conditions:1. For each i ∈ N, xy^i z ∈ L(M ).2. |y| > 0 (i.e., y contains at least one character).3. |xy| ≤ p (i.e., the string xy has at most p characters).
Use the pumping lemma to show the following language is not regular (HINT: Use proof by contradictionto assume the language is regular and apply the pumping lemma to the language):L = {0^k1^k | k ∈ N}
A prefix of length ℓ of some word w are the first ℓ characters (in order) of w.1. Construct a context-free grammar for the language: L = {w ∈ {a, b}∗ | every prefix of w has at least as many a’s as b’s}2. Explain why every word generated by your context-free grammar (in Part 1) is contained in L. Then,prove via induction that every w ∈ L is produced by your context-free grammar.
Consider a simplified version of American football where on any possession ateam can earn 0, 3 or 7 points. What is the smallest number n0 of points such that for all n ≥ n0 and n ∈ Na team could earn n points. You must prove that your answer is correct via induction (HINT: Don’t forgetto show that n0 is the smallest number above which any number of points is reachable).
Chapter 5 Solutions
Mathematics All Around (6th Edition)
Ch. 5.1 - Write the Egyptian numerals using Hindu-Arabic...Ch. 5.1 - Write the Egyptian numerals using Hindu-Arabic...Ch. 5.1 - Prob. 3ECh. 5.1 - Write the Egyptian numerals using Hindu-Arabic...Ch. 5.1 - Write each Hindu-Arabic numeral using Egyptian...Ch. 5.1 - Write each Hindu-Arabic numeral using Egyptian...Ch. 5.1 - Prob. 7ECh. 5.1 - Write each Hindu-Arabic numeral using Egyptian...Ch. 5.1 - Prob. 9ECh. 5.1 - Perform each of the following addition problems...
Ch. 5.1 - Prob. 11ECh. 5.1 - Perform each of the following addition problems...Ch. 5.1 - Perform each of the following subtraction problems...Ch. 5.1 - Perform each of the following subtraction problems...Ch. 5.1 - Perform each of the following subtraction problems...Ch. 5.1 - Perform each of the following subtraction problems...Ch. 5.1 - Prob. 17ECh. 5.1 - Use the Egyptian method of doubling to calculate...Ch. 5.1 - Prob. 19ECh. 5.1 - Use the Egyptian method of doubling to calculate...Ch. 5.1 - Prob. 21ECh. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Write each Roman numeral using Hindu-Arabic...Ch. 5.1 - Prob. 31ECh. 5.1 - Write each numeral in Roman notation There may be...Ch. 5.1 - Prob. 33ECh. 5.1 - Write each numeral in Roman notation There may be...Ch. 5.1 - Prob. 35ECh. 5.1 - Write each numeral in Roman notation There may be...Ch. 5.1 - Prob. 37ECh. 5.1 - Write each numeral in Roman notation There may be...Ch. 5.1 - Write each Chinese numeral as a Hindu-Arabic...Ch. 5.1 - Write each Chinese numeral as a Hindu-Arabic...Ch. 5.1 - Write each Chinese numeral as a Hindu-Arabic...Ch. 5.1 - Write each Chinese numeral as a Hindu-Arabic...Ch. 5.1 - Write each Chinese numeral as a Hindu-Arabic...Ch. 5.1 - Write each Chinese numeral as a Hindu-Arabic...Ch. 5.1 - Write each numeral using Chinese numerals. 495Ch. 5.1 - Write each numeral using Chinese numerals. 726Ch. 5.1 - Write each numeral using Chinese numerals. 2,805Ch. 5.1 - Write each numeral using Chinese numerals. 3,926Ch. 5.1 - Write each numeral using Chinese numerals. 9,846Ch. 5.1 - Write each numeral using Chinese numerals. 8,054Ch. 5.1 - The Great Pyramid at Giza was completed in . Write...Ch. 5.1 - Cheops, the builder of the Great Pyramid at Giza,...Ch. 5.1 - An Egyptian merchant has a warehouse that contains...Ch. 5.1 - An ancient Egyptian merchant had on hand bushels...Ch. 5.1 - Using Egyptian notation, the number 100,...Ch. 5.1 - Using Egyptian notation, the number 100,...Ch. 5.1 - Using Egyptian notation, the number 100,...Ch. 5.1 - Using Egyptian notation, the number 100,...Ch. 5.1 - The emperor Aurelius Constantine, who lived from...Ch. 5.1 - By 285ad, the Roman Empire had become so vast that...Ch. 5.1 - Frequently, Roman numerals are used today in movie...Ch. 5.1 - Prob. 62ECh. 5.1 - Prob. 63ECh. 5.1 - Frequently, Roman numerals are used today in movie...Ch. 5.1 - The counting boards In Exercises 6568 show...Ch. 5.1 - The counting boards In Exercises 6568 show...Ch. 5.1 - The counting boards In Exercises 6568 show...Ch. 5.1 - The counting boards In Exercises 6568 show...Ch. 5.1 - The oldest discovery of Chinese written numerals...Ch. 5.1 - When Marco Polo visited China in 1274, he was...Ch. 5.1 - Explain two advantages of the Roman numeration...Ch. 5.1 - The Roman numeration system has symbols for 5,50,...Ch. 5.1 - The traditional Chinese numeration system had no...Ch. 5.1 - Research the Ionic Greek numeration system, which...Ch. 5.1 - In the Egyptian numeration system, whenever we...Ch. 5.1 - Suppose that Egyptian numeration was based on 5...Ch. 5.1 - Invent an Egyptian type of numeration system using...Ch. 5.1 - Write the number 1,999 in Roman numerals in as...Ch. 5.1 - Egyptian mathematics had a unique way of writing...Ch. 5.1 - Egyptian mathematics had a unique way of writing...Ch. 5.1 - Egyptian mathematics had a unique way of writing...Ch. 5.1 - Egyptian mathematics had a unique way of writing...Ch. 5.2 - Write the following Babylonian numerals as...Ch. 5.2 - Write the following Babylonian numerals as...Ch. 5.2 - Write the following Babylonian numerals as...Ch. 5.2 - Write the following Babylonian numerals as...Ch. 5.2 - Write each number using Babylonian notation. 8,235Ch. 5.2 - Write each number using Babylonian notation. 7,331Ch. 5.2 - Write each number using Babylonian notation....Ch. 5.2 - Write each number using Babylonian notation....Ch. 5.2 - Write each number using Babylonian notation....Ch. 5.2 - Write each number using Babylonian notation....Ch. 5.2 - Write each number using Babylonian notation....Ch. 5.2 - Write each number using Babylonian notation....Ch. 5.2 - Translate each of the following Mayan numerals to...Ch. 5.2 - Translate each of the following Mayan numerals to...Ch. 5.2 - Translate each of the following Mayan numerals to...Ch. 5.2 - Translate each of the following Mayan numerals to...Ch. 5.2 - Write each number using Mayan notation. 17Ch. 5.2 - Write each number using Mayan notation. 48Ch. 5.2 - Prob. 19ECh. 5.2 - Prob. 20ECh. 5.2 - Prob. 21ECh. 5.2 - Prob. 22ECh. 5.2 - Prob. 23ECh. 5.2 - Prob. 24ECh. 5.2 - Prob. 25ECh. 5.2 - Prob. 26ECh. 5.2 - Prob. 27ECh. 5.2 - Prob. 28ECh. 5.2 - Prob. 29ECh. 5.2 - Prob. 30ECh. 5.2 - Prob. 31ECh. 5.2 - Prob. 32ECh. 5.2 - Prob. 33ECh. 5.2 - Prob. 34ECh. 5.2 - Prob. 35ECh. 5.2 - Prob. 36ECh. 5.2 - Prob. 37ECh. 5.2 - Prob. 38ECh. 5.2 - Prob. 39ECh. 5.2 - Prob. 40ECh. 5.2 - Prob. 41ECh. 5.2 - Prob. 42ECh. 5.2 - Prob. 43ECh. 5.2 - Prob. 44ECh. 5.2 - Prob. 45ECh. 5.2 - Prob. 46ECh. 5.2 - Prob. 47ECh. 5.2 - Prob. 48ECh. 5.2 - Prob. 49ECh. 5.2 - Prob. 50ECh. 5.2 - Prob. 51ECh. 5.2 - Prob. 52ECh. 5.2 - Prob. 53ECh. 5.2 - Prob. 54ECh. 5.2 - Prob. 55ECh. 5.2 - Prob. 56ECh. 5.2 - Prob. 57ECh. 5.2 - Prob. 58ECh. 5.2 - Prob. 59ECh. 5.2 - Prob. 60ECh. 5.2 - Prob. 61ECh. 5.2 - Prob. 62ECh. 5.2 - Prob. 63ECh. 5.2 - Prob. 64ECh. 5.2 - Prob. 65ECh. 5.2 - Prob. 66ECh. 5.2 - Prob. 67ECh. 5.2 - Prob. 68ECh. 5.2 - Prob. 69ECh. 5.2 - Prob. 70ECh. 5.2 - Prob. 71ECh. 5.2 - Prob. 72ECh. 5.2 - Prob. 73ECh. 5.2 - Prob. 74ECh. 5.2 - Prob. 75ECh. 5.2 - Prob. 76ECh. 5.2 - Prob. 77ECh. 5.2 - Prob. 78ECh. 5.2 - Prob. 79ECh. 5.2 - Prob. 80ECh. 5.3 - Prob. 1ECh. 5.3 - Prob. 2ECh. 5.3 - Prob. 3ECh. 5.3 - Prob. 4ECh. 5.3 - Prob. 5ECh. 5.3 - Prob. 6ECh. 5.3 - Prob. 7ECh. 5.3 - Prob. 8ECh. 5.3 - Prob. 9ECh. 5.3 - Prob. 10ECh. 5.3 - Prob. 11ECh. 5.3 - Prob. 12ECh. 5.3 - Prob. 13ECh. 5.3 - Prob. 14ECh. 5.3 - Prob. 15ECh. 5.3 - Prob. 16ECh. 5.3 - Prob. 17ECh. 5.3 - Prob. 18ECh. 5.3 - Prob. 19ECh. 5.3 - Prob. 20ECh. 5.3 - Prob. 21ECh. 5.3 - Prob. 22ECh. 5.3 - Prob. 23ECh. 5.3 - Prob. 24ECh. 5.3 - Prob. 25ECh. 5.3 - Prob. 26ECh. 5.3 - Prob. 27ECh. 5.3 - Prob. 28ECh. 5.3 - Prob. 29ECh. 5.3 - Prob. 30ECh. 5.3 - Prob. 31ECh. 5.3 - Prob. 32ECh. 5.3 - Prob. 33ECh. 5.3 - Prob. 34ECh. 5.3 - Prob. 35ECh. 5.3 - Prob. 36ECh. 5.3 - Prob. 37ECh. 5.3 - Prob. 38ECh. 5.3 - Prob. 39ECh. 5.3 - Prob. 40ECh. 5.3 - Prob. 41ECh. 5.3 - Prob. 42ECh. 5.3 - Prob. 43ECh. 5.3 - Prob. 44ECh. 5.3 - Prob. 45ECh. 5.3 - Prob. 46ECh. 5.3 - Prob. 47ECh. 5.3 - Prob. 48ECh. 5.3 - Prob. 49ECh. 5.3 - Prob. 50ECh. 5.3 - Prob. 51ECh. 5.3 - Prob. 52ECh. 5.3 - Prob. 53ECh. 5.3 - Prob. 54ECh. 5.3 - Prob. 55ECh. 5.3 - Prob. 56ECh. 5.3 - Prob. 57ECh. 5.3 - Prob. 58ECh. 5.3 - Prob. 59ECh. 5.3 - Prob. 60ECh. 5.3 - Prob. 61ECh. 5.3 - Prob. 62ECh. 5.3 - Prob. 63ECh. 5.3 - Prob. 64ECh. 5.3 - Prob. 65ECh. 5.3 - Prob. 66ECh. 5.3 - Prob. 67ECh. 5.3 - Prob. 68ECh. 5.3 - Prob. 69ECh. 5.3 - Prob. 70ECh. 5.3 - Prob. 71ECh. 5.3 - Prob. 72ECh. 5.3 - Prob. 73ECh. 5.3 - Prob. 74ECh. 5.3 - Prob. 75ECh. 5.3 - Prob. 76ECh. 5.3 - Prob. 77ECh. 5.3 - Prob. 78ECh. 5.3 - Prob. 79ECh. 5.3 - Prob. 80ECh. 5.3 - Prob. 81ECh. 5.3 - Prob. 82ECh. 5.3 - Prob. 83ECh. 5.3 - Prob. 84ECh. 5.3 - Prob. 85ECh. 5.3 - Prob. 86ECh. 5.3 - Prob. 87ECh. 5.3 - Prob. 88ECh. 5.3 - Prob. 89ECh. 5.3 - Prob. 90ECh. 5.3 - Prob. 91ECh. 5.3 - Prob. 92ECh. 5.3 - Prob. 93ECh. 5.3 - Prob. 94ECh. 5.3 - Prob. 95ECh. 5.3 - Prob. 96ECh. 5.3 - Prob. 97ECh. 5.3 - Prob. 98ECh. 5.3 - Prob. 99ECh. 5.3 - Prob. 100ECh. 5.3 - Prob. 101ECh. 5.3 - Prob. 102ECh. 5.4 - Prob. 1ECh. 5.4 - Prob. 2ECh. 5.4 - Prob. 3ECh. 5.4 - Prob. 4ECh. 5.4 - Prob. 5ECh. 5.4 - Prob. 6ECh. 5.4 - Prob. 7ECh. 5.4 - Prob. 8ECh. 5.4 - Prob. 9ECh. 5.4 - Prob. 10ECh. 5.4 - Prob. 11ECh. 5.4 - Prob. 12ECh. 5.4 - Prob. 13ECh. 5.4 - Prob. 14ECh. 5.4 - Prob. 15ECh. 5.4 - Prob. 16ECh. 5.4 - Prob. 17ECh. 5.4 - Prob. 18ECh. 5.4 - Prob. 19ECh. 5.4 - Prob. 20ECh. 5.4 - Prob. 21ECh. 5.4 - Prob. 22ECh. 5.4 - Prob. 23ECh. 5.4 - Prob. 24ECh. 5.4 - Prob. 25ECh. 5.4 - Prob. 26ECh. 5.4 - Prob. 27ECh. 5.4 - Prob. 28ECh. 5.4 - Prob. 29ECh. 5.4 - Prob. 30ECh. 5.4 - Prob. 31ECh. 5.4 - Prob. 32ECh. 5.4 - Prob. 33ECh. 5.4 - Prob. 34ECh. 5.4 - Prob. 35ECh. 5.4 - Prob. 36ECh. 5.4 - Prob. 37ECh. 5.4 - Prob. 38ECh. 5.4 - Prob. 39ECh. 5.4 - Prob. 40ECh. 5.4 - Prob. 41ECh. 5.4 - Prob. 42ECh. 5.4 - Prob. 43ECh. 5.4 - Prob. 44ECh. 5.4 - Prob. 45ECh. 5.4 - Prob. 46ECh. 5.4 - Prob. 47ECh. 5.4 - Prob. 48ECh. 5.4 - Prob. 49ECh. 5.4 - Prob. 50ECh. 5.4 - Prob. 51ECh. 5.4 - Prob. 52ECh. 5.4 - Prob. 53ECh. 5.4 - Prob. 54ECh. 5.4 - Prob. 55ECh. 5.4 - Prob. 56ECh. 5.4 - a. Why are check digits important? Give an...Ch. 5.4 - Prob. 58ECh. 5.4 - Prob. 59ECh. 5.4 - Prob. 60ECh. 5.4 - Prob. 61ECh. 5.4 - Prob. 62ECh. 5.4 - Prob. 63ECh. 5.4 - Challenge Yourself When we do usual division of...Ch. 5.4 - Prob. 65ECh. 5.CR - Prob. 1CRCh. 5.CR - Prob. 2CRCh. 5.CR - Prob. 3CRCh. 5.CR - Prob. 4CRCh. 5.CR - Prob. 5CRCh. 5.CR - Prob. 6CRCh. 5.CR - Prob. 7CRCh. 5.CR - Prob. 8CRCh. 5.CR - Prob. 9CRCh. 5.CR - Prob. 10CRCh. 5.CR - Prob. 11CRCh. 5.CR - Prob. 12CRCh. 5.CR - Prob. 13CRCh. 5.CR - Prob. 14CRCh. 5.CR - Prob. 15CRCh. 5.CR - Prob. 16CRCh. 5.CR - Prob. 17CRCh. 5.CR - Prob. 18CRCh. 5.CR - Prob. 19CRCh. 5.CR - Prob. 20CRCh. 5.CR - Prob. 21CRCh. 5.CR - Prob. 22CRCh. 5.CR - Prob. 23CRCh. 5.CT - Write 3,685 in Roman notation.Ch. 5.CT - Prob. 2CTCh. 5.CT - Write 2647 and A3E16 as base-10 numerals.Ch. 5.CT - Prob. 4CTCh. 5.CT - Prob. 5CTCh. 5.CT - Prob. 6CTCh. 5.CT - Prob. 7CTCh. 5.CT - Prob. 8CTCh. 5.CT - Prob. 9CTCh. 5.CT - Prob. 10CTCh. 5.CT - Prob. 11CTCh. 5.CT - Prob. 12CTCh. 5.CT - Prob. 13CTCh. 5.CT - Prob. 14CTCh. 5.CT - Prob. 15CTCh. 5.CT - Prob. 16CTCh. 5.CT - Prob. 17CTCh. 5.CT - Prob. 18CTCh. 5.CT - Prob. 19CTCh. 5.CT - Prob. 20CTCh. 5.CT - Prob. 21CTCh. 5.CT - Prob. 22CT
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