MA111: Introduction to Mathematical Reasoning

Welcome to MA111: Introduction to Mathematical Reasoning

Specific information about this course and its requirements can be found below. For more general information about taking Saylor Academy courses, including information about Community and Academic Codes of Conduct, please read the Student Handbook.

Course Description

Examine the language and practice of set theory, and the theory and practice of mathematical proof, with the purpose of getting from "doing mathematics" at an elementary, problem-solving level to "doing mathematics" at an advanced level.

Course Introduction

The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems. Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations. The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except for the first one, puzzles. There are several reasons for including puzzles. First and foremost, a challenging puzzle can be a microcosm of mathematical development. A great puzzle is like a laboratory for proving propositions. The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem. The mathematician "plays" with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler "plays" with the puzzle. Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing. At some stage, the puzzler (mathematician) develops sufficient sense of the structure, and only then can he begin to build the solution (prove the theorem). This multi-step process is perfectly mirrored in solving the KenKen problems this course presents. Some aspects of the solutions motivate ideas you will encounter later in the course. For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity.

This course includes the following units:

• Unit 1: Logic
• Unit 2: Sets
• Unit 3: Introduction to Number Theory
• Unit 4: Rational Numbers
• Unit 5: Mathematical Induction
• Unit 6: Relations and Functions
• Unit 7: Sets, Part II
• Unit 8: Combinatorics

Course Learning Outcomes

Upon successful completion of this course, you will be able to:

• read and dissect proofs of elementary propositions related to discrete mathematical objects such as integers, finite sets, graphs and relations, and functions;
• translate verbal statements into symbolic ones by using the elements of mathematical logic;
• determine when a proposed mathematical argument is logically correct;
• determine when a compound sentence is a tautology, a contradiction, or a contingency;
• translate riddles and other brainteasers into the language of predicates and propositions;
• solve problems related to place value, divisors, and remainders;
• use modular arithmetic to solve various equations, including quadratic equations in Z₆, Z₇, Z₁₁, and Diophantine equations;
• prove and use the salient characteristics of the rational, irrational, and real number systems to verify properties of various number systems;
• use mathematical induction to construct proofs of propositions about sets of positive integers;
• classify relations as being reflexive, symmetric, antisymmetric, transitive, a partial ordering, a total ordering, or an equivalence relation;
• determine if a relation is a function, and if so, whether or not it is a bijection;
• manipulate finite and infinite sets by using functions and set operations;
• determine if a set is finite, countable, or uncountable;
• use the properties of countable and uncountable sets in various situations;
• recognize some standard countable and uncountable sets; and
• determine and effectively use an appropriate counting tool to find the number of objects in a finite set.

Throughout this course, you will also see learning outcomes in each unit. You can use those learning outcomes to help organize your studies and gauge your progress.

Course Materials

The primary learning materials for this course are articles, lectures, and videos.

All course materials are free to access and can be found in each unit of the course. Pay close attention to the notes that accompany these course materials, as they will tell you what to focus on in each resource, and will help you to understand how the learning materials fit into the course as a whole. You can also see a list of all the learning materials in this course by clicking on Resources in the navigation bar.

Evaluation and Minimum Passing Score

Only the final exam is considered when awarding you a grade for this course. In order to pass this course, you will need to earn a 70% or higher on the final exam. Your score on the exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you may take it again as many times as you want, with a 7-day waiting period between each attempt. Once you have successfully passed the final exam you will be awarded a free Course Completion Certificate.

There are also end-of-unit assessments in this course. These are designed to help you study, and do not factor into your final course grade. You can take these as many times as you want until you understand the concepts and material covered. You can see all of these assessments by clicking on Quizzes in the course's navigation bar.

Tips for Success

MA111: Introduction to Mathematical Reasoning is a self-paced course, which means that you can decide when you will start and when you will complete the course. There is no instructor or an assigned schedule to follow. We estimate that the "average" student will take 112 hours to complete this course. We recommend that you work through the course at a pace that is comfortable for you and allows you to make regular progress. It's a good idea to also schedule your study time in advance and try as best as you can to stick to that schedule.

Learning new material can be challenging, so we've compiled a few study strategies to help you succeed:

• Take notes on the various terms, practices, and theories that you come across. This can help you put each concept into context, and will create a refresher that you can use as you study later on.
• As you work through the materials, take some time to test yourself on what you remember and how well you understand the concepts. Reflecting on what you've learned is important for your long-term memory, and will make you more likely to retain information over time.
• Although you may work through this course completely independently, you may find it helpful to connect with other Saylor students through the discussion forums. You may access the discussion forums at https://discourse.saylor.org.

Technical Requirements

This course is delivered entirely online. You will be required to have access to a computer or web-capable mobile device and have consistent access to the internet to either view or download the necessary course resources and to attempt any auto-graded course assessments and the final exam.

• To access the full course including assessments and the final exam, you will need to be logged into your Saylor Academy account and enrolled in the course. If you do not already have an account, you may create one for free here. Although you can access some of the course without logging in to your account, you should log in to maximize your course experience. For example, you cannot take assessments or track your progress unless you are logged in.

For additional guidance, check out Saylor Academy's FAQ.

Fees

This course is entirely free to enroll in and to access. Everything linked in the course, including textbooks, videos, webpages, and activities, is available for no charge. This course also contains a free final exam and course completion certificate.