Math in Society Mathematics for liberal arts majors Portland Community College Pilot Edition 0.3 Math in Society Mathematics for liberal arts majors Edition: Pilot 0.3 Website: http://spot.pcc.edu/~caralee/Math_105.html September 23, 2019 Portland Community College This book is a derivative of Math in Society, by David Lippman, et al, used under CC-BY-SA 3.0. Licensed by Portland Community College under CC-By-SA 3.0 This book was made possible by Open Oregon Educational Resources. i Attributions Project Lead: Cara Lee Contributing Authors: Jess Brooks Cara Lee Sonya Redmond Cindy Rochester-Gefre Thanks to Carlos Cantos and Virginia Somes for input and catching typos. Licensed by Portland Community College under CC BY-SA 3.0. Cover Image: Portland, Oregon Skyline from the Ross Island Bridge, by Visitor7, used under CC BY-SA 3.0 Unported, cropped. Chapter 1 is a derivative of Math in Society: Logic, by David Lippman and Morgan Chase, and Math in Society: Sets by David Lippman, used under CC-BY-SA 3.0. Sections 2.2-2.4 are a derivative of Math in Society: Finance, by David Lippman, used under CC-BY-SA 3.0. Sections 2.1 and 2.5 are original to Portland Community College. Section 2.5, Figure 1: Tax Buckets by John Chesbrough, used under CC-BY-ND-NC 4.0. Chapter 3 is a derivative of Math in Society: Describing Data and Statistics, by David Lippman, Jeff Eldridge and www.onlinestatbook.com, and www.onlinestatbook.com, by David M. Lane, et al, used under CC-BY-SA 3.0. Chapter 4 is a derivative of Math in Society: Probability, by David Lippman, used under CC-BY-SA 3.0. Technology Screenshots: All spreadsheet screenshots use Microsoft Excel under fair use. If you plan to redistribute this book, please consider whether your use is also fair use. GeoGebra screenshots are used for non-commercial use under https://www.geogebra.org/license#NonCommercialLicenseAgreement ii Notes We dedicate this book to our students May you have greater ease in paying for college and grow your proficiency and confidence in math. Word, PDF and Print Versions This book is available free online at http://spot.pcc.edu/~caralee/Math_105.html. There are Microsoft Word documents and PDF versions of each chapter. There is a PDF version of the required chapters (1-4) online and available at the bookstore for the cost of printing. Only the required chapters are in the printed version, to make the print version cheaper. This course includes one or more instructor choice topics which may be accessed either from the website above, or http://www.opentextbookstore.com/mathinsociety/index.html. Accessibility The word version of each chapter is accessible for use with screen readers. The accessible features include heading navigation, MathType and alternate text on all images. For truth tables and Venn diagrams, files with detailed figure descriptions and graphics optimized for tactile production can be found on the textbook website listed above. If you find anything that can improve the accessibility of this book, please email [email protected]. MyOpenMath Online homework problems are available for free at https://www.myopenmath.com/. Philosophy We emphasize technology, conceptual understanding and communication over rote calculation. However, some manual calculation is important to understand what the technology is doing. We emphasize readily available spreadsheets and GeoGebra throughout the text. Acknowledgements We would like to thank Amy Hofer of OpenOregon and the PCC OER steering committee. Thanks also to Kaela Parks and Michael Cantino of Disability Services for their expertise on accessibility and for producing the tactile model files. iii Table of Contents Chapter 1: Logic and Sets ....................................................................... 1 Section 1.1 The Language and Rules of Logic ..................................... 2 Section 1.2 Sets and Venn Diagrams ................................................ 11 Section 1.3 Describing and Critiquing Arguments ............................ 21 Section 1.4 Logical Fallacies .............................................................. 29 Chapter 2: Financial Math .................................................................... 33 Section 2.1 Introduction to Spreadsheets ........................................ 34 Section 2.2 Simple and Compound Interest ..................................... 38 Section 2.3 Savings Plans .................................................................. 51 Section 2.4 Loan Payments ............................................................... 61 Section 2.5 Income Taxes ................................................................. 73 Chapter 3: Statistics ............................................................................. 83 Section 3.1 Overview of the Statistical Process ................................ 84 Section 3.2 Describing Data .............................................................. 99 Section 3.3 Summary Statistics: Measures of Center ..................... 115 Section 3.4 Summary Statistics: Measures of Variation ................. 126 Chapter 4: Probability ........................................................................ 147 Section 4.1 Contingency Tables ...................................................... 148 Section 4.2 Theoretical Probability ................................................. 159 Section 4.3 Expected Value ............................................................ 171 iv Chapter 1: Logic and Sets Student Outcomes for this Chapter Section 1.1: The Language and Rules of Logic Students will be able to: Identify propositions Compose and interpret the negation of a statement Use logical connectors (and/or) and conditional statements (if, then) Use truth tables to find truth values of basic and complex statements Section 1.2: Sets and Venn Diagrams Students will be able to: Use set notation and understand the null set Determine the universal set for a given context Use Venn diagrams and set notation to illustrate the intersection, union and complements of sets Illustrate disjoint sets, subsets and overlapping sets with diagrams Use Venn diagrams and problem-solving strategies to solve logic problems Section 1.3: Describing and Critiquing Arguments. Students will be able to: Understand the structure of logical arguments by identifying the premise(s) and conclusion. Distinguish between inductive and deductive arguments Make a set diagram to evaluate deductive arguments Determine whether a deductive argument is valid and/or sound Section 1.4: Logical Fallacies. Students will be able to: Identify common logical fallacies and their use in arguments Chapter 1 is a derivative of Math in Society: Logic, by David Lippman and Morgan Chase, and Math in Society: Sets by David Lippman, used under CC-BY-SA 3.0. Licensed by Portland Community College under CC-By-SA 3.0. Chapter 1: Logic and Sets Section 1.1 The Language and Rules of Logic Logic Logic is the study of reasoning. Our goal in this chapter is to examine arguments to determine their validity and soundness. In this section we will look at propositions and logical connectors that are the building blocks of arguments. We will also use truth tables to help us examine complex statements. Propositions A proposition is a complete sentence that is either true or false. Opinions can be propositions, but questions or phrases cannot. Example 1: Which of the following are propositions? a. I am reading a math book. b. Math is fun! c. Do you like turtles? d. My cat The first and second items are propositions. The third one is a question and the fourth is a phrase, so they are not. We are not concerned right now about whether a statement is true or false. We will come back to that later when we examine full arguments. Arguments are made of one or more propositions (called premises), along with a conclusion. Propositions may be negated, or combined with connectors like “and”, and “or”. Let’s take a closer look at how these negations and logical connectors are used to create more complex statements. Negation (not) One way to change a proposition is to use its negation, or opposite meaning. We often use the word “not” to negate a statement. Example 2: Write the negation of the following propositions. a. I am reading a math book. Negation: I am not reading a math book. b. Math is fun! Negation: Math is not fun! c. The sky is not green. Negation: The sky is green (or not not green). d. Cars have wheels Negation: Cars do not have wheels. Multiple Negations It is possible to use more than one negation in a statement. If you’ve ever said something like, “I can’t not go,” you are really saying you must go. It’s a lot like multiplying two negative numbers which gives a positive result. 2 1.1 The Language and Rules of Logic In the media and in ballot measures we often see multiple negations and it can be confusing to figure out what a statement means. Example 3: Read the statement to determine the outcome of a yes vote. “Vote for this measure to repeal the ban on plastic bags.” If you said that a yes vote would enable plastic bag usage, you are correct. The ban stopped plastic bag usage, so to repeal the ban would allow it again. This measure has a double negation and is also not very good for the environment. Example 4: Read the statement to determine the outcome on mandatory minimum sentencing. “The bill that overturned the ban on mandatory minimum sentencing was vetoed.” In this case mandatory minimum sentencing would not be allowed. The ban would stop it, and the bill to