Explanation Procedure: The properties that must be satisfied by a set to be a group for modulo addition are: 1. Closure property: The sum of any pair of elements must also be a member of the original set. 2. Associative property: For all elements, a , b and c belonging to the set, the following equation must be satisfied. ( a + b ) + c ≡ a + ( b + c ) 3. Identity property: The additive identity element 0 must be in the set such that the sum of any element in the set and the identity is the same element. 4. Inverse property: For each element a that belongs to the set, there must be a unique element a’ belonging to the set such that: a + a ' ≡ a ' + a ≡ 0 Here, a ’ is said to be the inverse of the element a . Calculation: Consider the set { 0 , 1 , 2 , 3 } . It has to be determined whether the provided set is a group for addition modulo 4. So check whether all the properties are satisfied. 1. Closure property: For the provided set of elements modulo 4, the sum of any pair of elements is also a member of the set. For example, 1 + 2 = 3 ≡ 3 , ( mod 4 ) 2 + 3 = 5 ≡ 1 , ( mod 4 ) So, the set is closed with respect to addition. 2. Associative property: The associative property is satisfied for the provided set of elements. That is, ( a + b ) + c ≡ a + ( b + c ) for all elements. So, consider three elements 1, 2, 3

Nature of Mathematics (MindTap Course List)

13th Edition

ISBN: 9781133947257

Author: karl J. smith

Publisher: Cengage Learning

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

Chapter 5.7, Problem 39PS

To determine

To find:

Whether the set {0,1,2,3} is a group for addition modulo 4.

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Chapter 5.7, Problem 38PS

Chapter 5.7, Problem 40PS

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Chapter 5 Solutions

Nature of Mathematics (MindTap Course List)

Ch. 5.1 - Level 1 IN YOUR OWN WORDS Explain each of the...Ch. 5.1 - Prob. 2PS Ch. 5.1 - Prob. 3PS Ch. 5.1 - Prob. 4PS Ch. 5.1 - Prob. 5PS Ch. 5.1 - Prob. 6PS Ch. 5.1 - Prob. 7PS Ch. 5.1 - Prob. 8PS Ch. 5.1 - Prob. 9PS Ch. 5.1 - Prob. 10PS

Ch. 5.1 - Prob. 11PS Ch. 5.1 - Prob. 12PS Ch. 5.1 - Prob. 13PS Ch. 5.1 - Prob. 14PS Ch. 5.1 - Prob. 15PS Ch. 5.1 - Prob. 16PS Ch. 5.1 - Prob. 17PS Ch. 5.1 - Prob. 18PS Ch. 5.1 - Prob. 19PS Ch. 5.1 - Prob. 20PS Ch. 5.1 - Prob. 21PS Ch. 5.1 - Level 1 In Problems 15-26, classify each as an...Ch. 5.1 - Prob. 23PS Ch. 5.1 - Prob. 24PS Ch. 5.1 - Prob. 25PS Ch. 5.1 - Prob. 26PS Ch. 5.1 - Prob. 27PS Ch. 5.1 - Prob. 28PS Ch. 5.1 - Prob. 29PS Ch. 5.1 - Level 1 Think of three nonassociative word triples...Ch. 5.1 - Prob. 31PS Ch. 5.1 - Prob. 32PS Ch. 5.1 - Prob. 33PS Ch. 5.1 - Prob. 34PS Ch. 5.1 - Prob. 35PS Ch. 5.1 - Prob. 36PS Ch. 5.1 - Prob. 37PS Ch. 5.1 - Prob. 38PS Ch. 5.1 - Prob. 39PS Ch. 5.1 - Prob. 40PS Ch. 5.1 - Prob. 41PS Ch. 5.1 - Prob. 42PS Ch. 5.1 - Prob. 43PS Ch. 5.1 - Prob. 44PS Ch. 5.1 - Prob. 45PS Ch. 5.1 - Prob. 46PS Ch. 5.1 - Prob. 47PS Ch. 5.1 - Prob. 48PS Ch. 5.1 - Prob. 49PS Ch. 5.1 - Prob. 50PS Ch. 5.1 - Prob. 51PS Ch. 5.1 - Prob. 52PS Ch. 5.1 - Prob. 53PS Ch. 5.1 - Prob. 54PS Ch. 5.1 - Prob. 55PS Ch. 5.1 - Problem Solving Level 3 Consider a soldier facing...Ch. 5.1 - Prob. 57PS Ch. 5.1 - Prob. 58PS Ch. 5.1 - Prob. 59PS Ch. 5.1 - Problem Solving Level 3 60. The Vanishing...Ch. 5.2 - IN YOUR OWN WORDS What is a prime number?Ch. 5.2 - Prob. 2PS Ch. 5.2 - Prob. 3PS Ch. 5.2 - Prob. 4PS Ch. 5.2 - Prob. 5PS Ch. 5.2 - Prob. 6PS Ch. 5.2 - Prob. 7PS Ch. 5.2 - Prob. 8PS Ch. 5.2 - Prob. 9PS Ch. 5.2 - Prob. 10PS Ch. 5.2 - Prob. 11PS Ch. 5.2 - Prob. 12PS Ch. 5.2 - Prob. 13PS Ch. 5.2 - Prob. 14PS Ch. 5.2 - Find all the prime numbers less than or equal to...Ch. 5.2 - Determine the largest prime you need to consider...Ch. 5.2 - Prob. 17PS Ch. 5.2 - Prob. 18PS Ch. 5.2 - Prob. 19PS Ch. 5.2 - Prob. 20PS Ch. 5.2 - Prob. 21PS Ch. 5.2 - Prob. 22PS Ch. 5.2 - Prob. 23PS Ch. 5.2 - Prob. 24PS Ch. 5.2 - Prob. 25PS Ch. 5.2 - Prob. 26PS Ch. 5.2 - Prob. 27PS Ch. 5.2 - Prob. 28PS Ch. 5.2 - Prob. 29PS Ch. 5.2 - Prob. 30PS Ch. 5.2 - Prob. 31PS Ch. 5.2 - Prob. 32PS Ch. 5.2 - Prob. 33PS Ch. 5.2 - Prob. 34PS Ch. 5.2 - Prob. 35PS Ch. 5.2 - Prob. 36PS Ch. 5.2 - Prob. 37PS Ch. 5.2 - Prob. 38PS Ch. 5.2 - Prob. 39PS Ch. 5.2 - Two movie theaters, UAI and UAII, start their...Ch. 5.2 - Prob. 41PS Ch. 5.2 - Shannon is planning a party and wants to give each...Ch. 5.2 - Boxes that are 12 inches tall are stacked next to...Ch. 5.2 - Hot dogs come in packages of 10, while buns come...Ch. 5.2 - IN YOUR OWN WORDS We used a sieve of Eratosthenes...Ch. 5.2 - Prob. 46PS Ch. 5.2 - Prob. 47PS Ch. 5.2 - Prob. 48PS Ch. 5.2 - Prob. 49PS Ch. 5.2 - Prob. 50PS Ch. 5.2 - Prob. 51PS Ch. 5.2 - Prob. 52PS Ch. 5.2 - Prob. 53PS Ch. 5.2 - Prob. 54PS Ch. 5.2 - Prob. 55PS Ch. 5.2 - What is the smallest natural number that is...Ch. 5.2 - Prob. 57PS Ch. 5.2 - Prob. 58PS Ch. 5.2 - Prob. 59PS Ch. 5.2 - Prob. 60PS Ch. 5.3 - Prob. 1PS Ch. 5.3 - Prob. 2PS Ch. 5.3 - Prob. 3PS Ch. 5.3 - Prob. 4PS Ch. 5.3 - Prob. 5PS Ch. 5.3 - Prob. 6PS Ch. 5.3 - Prob. 7PS Ch. 5.3 - Prob. 8PS Ch. 5.3 - Prob. 9PS Ch. 5.3 - Prob. 10PS Ch. 5.3 - Prob. 11PS Ch. 5.3 - Prob. 12PS Ch. 5.3 - Prob. 13PS Ch. 5.3 - Prob. 14PS Ch. 5.3 - Prob. 15PS Ch. 5.3 - Prob. 16PS Ch. 5.3 - Prob. 17PS Ch. 5.3 - Prob. 18PS Ch. 5.3 - Prob. 19PS Ch. 5.3 - Simplify the expressions in Problems 9-37. a....Ch. 5.3 - Prob. 21PS Ch. 5.3 - Prob. 22PS Ch. 5.3 - Prob. 23PS Ch. 5.3 - Prob. 24PS Ch. 5.3 - Prob. 25PS Ch. 5.3 - Prob. 26PS Ch. 5.3 - Prob. 27PS Ch. 5.3 - Prob. 28PS Ch. 5.3 - Prob. 29PS Ch. 5.3 - Prob. 30PS Ch. 5.3 - Prob. 31PS Ch. 5.3 - Prob. 32PS Ch. 5.3 - Prob. 33PS Ch. 5.3 - Prob. 34PS Ch. 5.3 - Prob. 35PS Ch. 5.3 - Prob. 36PS Ch. 5.3 - Prob. 37PS Ch. 5.3 - Prob. 38PS Ch. 5.3 - Prob. 39PS Ch. 5.3 - Prob. 40PS Ch. 5.3 - Prob. 41PS Ch. 5.3 - Prob. 42PS Ch. 5.3 - Prob. 43PS Ch. 5.3 - Prob. 44PS Ch. 5.3 - Prob. 45PS Ch. 5.3 - Prob. 46PS Ch. 5.3 - Prob. 47PS Ch. 5.3 - Prob. 48PS Ch. 5.3 - Prob. 49PS Ch. 5.3 - Prob. 50PS Ch. 5.3 - Prob. 51PS Ch. 5.3 - Prob. 52PS Ch. 5.3 - Perform the indicated operations. Let k be a...Ch. 5.3 - Perform the indicated operations. Let k be a...Ch. 5.3 - a. State the commutative property. b. Is ...Ch. 5.3 - Prob. 56PS Ch. 5.3 - Prob. 57PS Ch. 5.3 - Prob. 58PS Ch. 5.3 - Multiply 1,234,5679,999,999 a. using a calculator...Ch. 5.3 - Prob. 60PS Ch. 5.4 - IN YOUR OWN WORDS What does it mean for a fraction...Ch. 5.4 - IN YOUR OWN WORDS Describe the process for...Ch. 5.4 - Prob. 3PS Ch. 5.4 - Prob. 4PS Ch. 5.4 - IN YOUR OWN WORDS Use algebra to show where the...Ch. 5.4 - IN YOUR OWN WORDS Use algebra to show where the...Ch. 5.4 - Completely reduce the fractions in Problems 7-18....Ch. 5.4 - Completely reduce the fractions in Problems 7-18....Ch. 5.4 - Completely reduce the fractions in Problems 7-18....Ch. 5.4 - Completely reduce the fractions in Problems 7-18....Ch. 5.4 - Completely reduce the fractions in problem 7-18....Ch. 5.4 - Completely reduce the fractions in problem 7-18....Ch. 5.4 - Completely reduce the fractions in problem 7-18....Ch. 5.4 - Completely reduce the fractions in problem 7-18....Ch. 5.4 - Completely reduce the fractions in problem 7-18....Ch. 5.4 - Completely reduce the fractions in problem 7-18....Ch. 5.4 - Completely reduce the fractions in problem 7-18....Ch. 5.4 - Completely reduce the fractions in problem 7-18....Ch. 5.4 - Perform the indicated operations in problems...Ch. 5.4 - Perform the indicated operations in problems...Ch. 5.4 - Perform the indicated operations in problems...Ch. 5.4 - Perform the indicated operations in problems...Ch. 5.4 - Perform the indicated operations in problems...Ch. 5.4 - Prob. 24PS Ch. 5.4 - Prob. 25PS Ch. 5.4 - Prob. 26PS Ch. 5.4 - Prob. 27PS Ch. 5.4 - Prob. 28PS Ch. 5.4 - Prob. 29PS Ch. 5.4 - Prob. 30PS Ch. 5.4 - Prob. 31PS Ch. 5.4 - Prob. 32PS Ch. 5.4 - Prob. 33PS Ch. 5.4 - Prob. 34PS Ch. 5.4 - Prob. 35PS Ch. 5.4 - Prob. 36PS Ch. 5.4 - Prob. 37PS Ch. 5.4 - Prob. 38PS Ch. 5.4 - Prob. 39PS Ch. 5.4 - Prob. 40PS Ch. 5.4 - Prob. 41PS Ch. 5.4 - Prob. 42PS Ch. 5.4 - Prob. 43PS Ch. 5.4 - Prob. 44PS Ch. 5.4 - Perform the indicated operations in problems...Ch. 5.4 - Perform the indicated operations in problems...Ch. 5.4 - Prob. 47PS Ch. 5.4 - Prob. 48PS Ch. 5.4 - Prob. 49PS Ch. 5.4 - Prob. 50PS Ch. 5.4 - Prob. 51PS Ch. 5.4 - Prob. 52PS Ch. 5.4 - Prob. 53PS Ch. 5.4 - Prob. 54PS Ch. 5.4 - Prob. 55PS Ch. 5.4 - Prob. 56PS Ch. 5.4 - Prob. 57PS Ch. 5.4 - Prob. 58PS Ch. 5.4 - Prob. 59PS Ch. 5.4 - Prob. 60PS Ch. 5.5 - Prob. 1PS Ch. 5.5 - Prob. 2PS Ch. 5.5 - Prob. 3PS Ch. 5.5 - Prob. 4PS Ch. 5.5 - Prob. 5PS Ch. 5.5 - Prob. 6PS Ch. 5.5 - Prob. 7PS Ch. 5.5 - Prob. 8PS Ch. 5.5 - Prob. 9PS Ch. 5.5 - Prob. 10PS Ch. 5.5 - Prob. 11PS Ch. 5.5 - Prob. 12PS Ch. 5.5 - Prob. 13PS Ch. 5.5 - Prob. 14PS Ch. 5.5 - Prob. 15PS Ch. 5.5 - Prob. 16PS Ch. 5.5 - Prob. 17PS Ch. 5.5 - Prob. 18PS Ch. 5.5 - Prob. 19PS Ch. 5.5 - Prob. 20PS Ch. 5.5 - Prob. 21PS Ch. 5.5 - Prob. 22PS Ch. 5.5 - Prob. 23PS Ch. 5.5 - Prob. 24PS Ch. 5.5 - Prob. 25PS Ch. 5.5 - Prob. 26PS Ch. 5.5 - Prob. 27PS Ch. 5.5 - Prob. 28PS Ch. 5.5 - Prob. 29PS Ch. 5.5 - Prob. 30PS Ch. 5.5 - Prob. 31PS Ch. 5.5 - Prob. 32PS Ch. 5.5 - Prob. 33PS Ch. 5.5 - Prob. 34PS Ch. 5.5 - Prob. 35PS Ch. 5.5 - Prob. 36PS Ch. 5.5 - Prob. 37PS Ch. 5.5 - Prob. 38PS Ch. 5.5 - If a carpenter wants to make sure that the corner...Ch. 5.5 - Prob. 40PS Ch. 5.5 - An antenna is to be erected and held by guy wires....Ch. 5.5 - Prob. 42PS Ch. 5.5 - Prob. 43PS Ch. 5.5 - Prob. 44PS Ch. 5.5 - Prob. 45PS Ch. 5.5 - Prob. 46PS Ch. 5.5 - Prob. 47PS Ch. 5.5 - Prob. 48PS Ch. 5.5 - Prob. 49PS Ch. 5.5 - Prob. 50PS Ch. 5.5 - Prob. 51PS Ch. 5.5 - Prob. 52PS Ch. 5.5 - Prob. 53PS Ch. 5.5 - Prob. 54PS Ch. 5.5 - Prob. 55PS Ch. 5.5 - Prob. 56PS Ch. 5.5 - Prob. 57PS Ch. 5.5 - Prob. 58PS Ch. 5.5 - Prob. 59PS Ch. 5.5 - Prob. 60PS Ch. 5.6 - Prob. 1PS Ch. 5.6 - Prob. 2PS Ch. 5.6 - Prob. 3PS Ch. 5.6 - Prob. 4PS Ch. 5.6 - Prob. 5PS Ch. 5.6 - Prob. 6PS Ch. 5.6 - Prob. 7PS Ch. 5.6 - Tell whether each number is an element of N a...Ch. 5.6 - Prob. 9PS Ch. 5.6 - Prob. 10PS Ch. 5.6 - Express each of the numbers in Problems 9-12 as a...Ch. 5.6 - Prob. 12PS Ch. 5.6 - Prob. 13PS Ch. 5.6 - Prob. 14PS Ch. 5.6 - Prob. 15PS Ch. 5.6 - Prob. 16PS Ch. 5.6 - Prob. 17PS Ch. 5.6 - Prob. 18PS Ch. 5.6 - Prob. 19PS Ch. 5.6 - Prob. 20PS Ch. 5.6 - Prob. 21PS Ch. 5.6 - Prob. 22PS Ch. 5.6 - Prob. 23PS Ch. 5.6 - Prob. 24PS Ch. 5.6 - Prob. 25PS Ch. 5.6 - Carry out the operations with decimal forms in...Ch. 5.6 - Prob. 27PS Ch. 5.6 - Prob. 28PS Ch. 5.6 - Prob. 29PS Ch. 5.6 - Prob. 30PS Ch. 5.6 - Prob. 31PS Ch. 5.6 - Prob. 32PS Ch. 5.6 - Prob. 33PS Ch. 5.6 - Prob. 34PS Ch. 5.6 - Prob. 35PS Ch. 5.6 - Prob. 36PS Ch. 5.6 - Prob. 37PS Ch. 5.6 - Prob. 38PS Ch. 5.6 - Prob. 39PS Ch. 5.6 - Prob. 40PS Ch. 5.6 - Prob. 41PS Ch. 5.6 - Prob. 42PS Ch. 5.6 - Prob. 43PS Ch. 5.6 - Prob. 44PS Ch. 5.6 - Prob. 45PS Ch. 5.6 - Prob. 46PS Ch. 5.6 - Prob. 47PS Ch. 5.6 - Prob. 48PS Ch. 5.6 - Prob. 49PS Ch. 5.6 - Prob. 50PS Ch. 5.6 - Prob. 51PS Ch. 5.6 - Prob. 52PS Ch. 5.6 - Prob. 53PS Ch. 5.6 - Prob. 54PS Ch. 5.6 - Prob. 55PS Ch. 5.6 - Let be an arbitrary operation in Problems 52-59....Ch. 5.6 - Let be an arbitrary operation in Problems 52-59....Ch. 5.6 - Prob. 58PS Ch. 5.6 - Prob. 59PS Ch. 5.6 - Prob. 60PS Ch. 5.7 - Level 1. IN YOUR OWN WORDS. What do we mean by...Ch. 5.7 - Prob. 2PS Ch. 5.7 - Prob. 3PS Ch. 5.7 - Prob. 4PS Ch. 5.7 - Prob. 5PS Ch. 5.7 - Prob. 6PS Ch. 5.7 - Level 1. Perform the indicated operations in...Ch. 5.7 - Prob. 8PS Ch. 5.7 - Prob. 9PS Ch. 5.7 - Prob. 10PS Ch. 5.7 - Prob. 11PS Ch. 5.7 - Prob. 12PS Ch. 5.7 - Prob. 13PS Ch. 5.7 - Prob. 14PS Ch. 5.7 - Prob. 15PS Ch. 5.7 - Prob. 16PS Ch. 5.7 - Prob. 17PS Ch. 5.7 - Prob. 18PS Ch. 5.7 - Prob. 19PS Ch. 5.7 - Prob. 20PS Ch. 5.7 - Prob. 21PS Ch. 5.7 - Prob. 22PS Ch. 5.7 - Prob. 23PS Ch. 5.7 - Prob. 24PS Ch. 5.7 - Prob. 25PS Ch. 5.7 - Prob. 26PS Ch. 5.7 - Prob. 27PS Ch. 5.7 - Prob. 28PS Ch. 5.7 - Prob. 29PS Ch. 5.7 - Prob. 30PS Ch. 5.7 - Prob. 31PS Ch. 5.7 - Prob. 32PS Ch. 5.7 - Prob. 33PS Ch. 5.7 - Prob. 34PS Ch. 5.7 - Prob. 35PS Ch. 5.7 - Level 2. If you know that your aunt in Problem 34...Ch. 5.7 - Prob. 37PS Ch. 5.7 - Prob. 38PS Ch. 5.7 - Prob. 39PS Ch. 5.7 - Prob. 40PS Ch. 5.7 - Prob. 41PS Ch. 5.7 - Prob. 42PS Ch. 5.7 - Prob. 43PS Ch. 5.7 - Prob. 44PS Ch. 5.7 - Prob. 45PS Ch. 5.7 - Prob. 46PS Ch. 5.7 - Prob. 47PS Ch. 5.7 - Prob. 48PS Ch. 5.7 - Prob. 49PS Ch. 5.7 - Prob. 50PS Ch. 5.7 - Prob. 51PS Ch. 5.7 - Prob. 52PS Ch. 5.7 - Prob. 53PS Ch. 5.7 - Prob. 54PS Ch. 5.7 - Prob. 55PS Ch. 5.7 - Prob. 56PS Ch. 5.7 - Prob. 57PS Ch. 5.7 - Prob. 58PS Ch. 5.7 - Prob. 59PS Ch. 5.7 - Prob. 60PS Ch. 5.8 - Prob. 1PS Ch. 5.8 - Prob. 2PS Ch. 5.8 - Prob. 3PS Ch. 5.8 - Prob. 4PS Ch. 5.8 - Prob. 5PS Ch. 5.8 - Prob. 6PS Ch. 5.8 - Prob. 7PS Ch. 5.8 - Prob. 8PS Ch. 5.8 - Prob. 9PS Ch. 5.8 - Number the letters of the alphabet from 1 to 26;...Ch. 5.8 - Prob. 11PS Ch. 5.8 - Prob. 12PS Ch. 5.8 - Prob. 13PS Ch. 5.8 - Prob. 14PS Ch. 5.8 - Prob. 15PS Ch. 5.8 - Prob. 16PS Ch. 5.8 - Prob. 17PS Ch. 5.8 - Prob. 18PS Ch. 5.8 - Prob. 19PS Ch. 5.8 - Prob. 20PS Ch. 5.8 - Prob. 21PS Ch. 5.8 - Prob. 22PS Ch. 5.8 - Prob. 23PS Ch. 5.8 - Prob. 24PS Ch. 5.8 - Prob. 25PS Ch. 5.8 - Prob. 26PS Ch. 5.8 - Prob. 27PS Ch. 5.8 - Prob. 28PS Ch. 5.8 - Prob. 29PS Ch. 5.8 - Prob. 30PS Ch. 5.8 - Prob. 31PS Ch. 5.8 - Give the decoding key for the encoding keys in...Ch. 5.8 - Prob. 33PS Ch. 5.8 - Prob. 34PS Ch. 5.8 - Prob. 35PS Ch. 5.8 - Prob. 36PS Ch. 5.8 - Prob. 37PS Ch. 5.8 - Prob. 38PS Ch. 5.8 - Prob. 39PS Ch. 5.8 - Use Figure 5.25 to encode or decode the messages...Ch. 5.8 - Prob. 41PS Ch. 5.8 - Prob. 42PS Ch. 5.8 - Prob. 43PS Ch. 5.8 - Prob. 44PS Ch. 5.8 - Prob. 45PS Ch. 5.8 - Prob. 46PS Ch. 5.8 - Prob. 47PS Ch. 5.8 - Prob. 48PS Ch. 5.8 - Prob. 51PS Ch. 5.8 - Prob. 52PS Ch. 5.8 - Prob. 53PS Ch. 5.8 - Prob. 54PS Ch. 5.8 - Prob. 55PS Ch. 5.8 - Prob. 56PS Ch. 5.8 - Prob. 57PS Ch. 5.8 - Prob. 58PS Ch. 5.8 - Prob. 59PS Ch. 5.8 - Prob. 60PS Ch. 5.CR - Prob. 1CR Ch. 5.CR - Prob. 2CR Ch. 5.CR - Prob. 3CR Ch. 5.CR - Prob. 4CR Ch. 5.CR - Reduce each fraction. If it is reduced, so state....Ch. 5.CR - Prob. 6CR Ch. 5.CR - Prob. 7CR Ch. 5.CR - Prob. 8CR Ch. 5.CR - Prob. 9CR Ch. 5.CR - Prob. 10CR Ch. 5.CR - Prob. 11CR Ch. 5.CR - Prob. 12CR Ch. 5.CR - Prob. 13CR Ch. 5.CR - Prob. 14CR Ch. 5.CR - Prob. 15CR Ch. 5.CR - Prob. 16CR Ch. 5.CR - Prob. 17CR Ch. 5.CR - Do You Weigh Too Much? In the United States, an...Ch. 5.CR - Prob. 19CR Ch. 5.CR - Prob. 20CR

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