Introduction to Mathematical Reasoning

Saylor Academy


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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 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).

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Numbers Theory


  • Unit 1: Logic

    In this unit, you will begin by considering various puzzles, including Ken-Ken and Sudoku.  You will learn the importance of tenacity in approaching mathematical problems including puzzles and brain teasers.  You will also learn why giving names to mathematical ideas will enable you to think more effectively about concepts that are built upon several ideas.  Then, you will learn that propositions are (English) sentences whose truth value can be established.  You will see examples of self-referencing sentences which are not propositions.  You will learn how to combine propositions to build compound ones and then how to determine the truth value of a compound proposition in terms of its component propositions.  Then, you will learn about predicates, which are functions from a collection of objects to a collection of propositions, and how to quantify predicates.  Finally, you will study several methods of proof including proof by contradiction, proof by complete enumeration, etc.

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  • Unit 2: Sets, Part I

    In this unit, you will explore the ideas of what is called 'naive set theory.'  Contrasted with 'axiomatic set theory,' naive set theory assumes that you already have an intuitive understanding of what it means to be a set.  You should mainly be concerned with how two or more given sets can be combined to build other sets and how the number of members (i.e. the cardinality) of such sets is related to the cardinality of the given sets.

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  • Unit 3: Introduction to Number Theory

    This unit is primarily concerned with the set of natural numbers N = {0, 1, 2, 3, . . .}.  The axiomatic approach to N will be postponed until the unit on recursion and mathematical induction.  This unit will help you understand the multiplication and additive structure of N.  This unit begins with integer representation: place value.  This fundamental idea enables you to completely understand the algorithms we learned in elementary school for addition, subtraction, multiplication, and division of multi-digit integers.  The beautiful idea in the Fusing Dots paper will enable you to develop a much deeper understanding of the representation of integers and other real numbers.  Then, you will learn about the multiplicative building blocks, the prime numbers.  The Fundamental Theorem of Arithmetic guarantees that every positive integer greater than 1 is a prime number or can be written as a product of prime numbers in essentially one way.  The Division Algorithm enables you to associate with each ordered pair of non-zero integers - a unique pair of integers - the quotient and the remainder.  Another important topic is modular arithmetic.  This arithmetic comes from an understanding of how remainders combine with one another under the operations of addition and multiplication.  Finally, the unit discusses the Euclidean Algorithm, which provides a method for solving certain equations over the integers.  Such equations with integer solutions are sometimes called Diophantine Equations.

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  • Unit 4: Rational Numbers

    In this unit, you will learn to prove some basic properties of rational numbers.  For example, the set of rational numbers is dense in the set of real numbers.  That means that strictly between any two real numbers, you can always find a rational number.  The distinction between a fraction and a rational number will also be discussed.  There is an easy way to tell whether a number given in decimal form is rational: if the digits of the representation regularly repeat in blocks, then the number is rational.  If this is the case, you can find a pair of integers whose quotient is the given decimal.  The unit discusses the mediant of a pair of rational fractions, and why the mediant does not depend on the values of its components, but instead on the way they are represented.

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  • Unit 5: Mathematical Induction

    In this unit, you will prove propositions about an infinite set of positive integers.  Mathematical induction is a technique used to formulate all such proofs.  The term recursion refers to a method of defining sequences of numbers, functions, and other objects.  The term mathematical induction refers to a method of proving properties of such recursively defined objects.

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  • Unit 6: Relations and Functions

    In this unit, you will learn about binary relations from a set A to a set B.  Some of these relations are functions from A to B.  Restricting our attention to relations from a set A to the set A, this unit discusses the properties of reflexivity(R), symmetry(S), anti-symmetry(A), and transitivity(T).  Relations that satisfy R, S, and T are called equivalence relations, and those satisfying R, A, and T are called partial orderings.

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  • Unit 7: Sets, Part II

    In this unit, you will study cardinality.  One startling realization is that not all infinite sets are the same size.  In fact, there are many different size infinite sets.  This can be made perfectly understandable to you at this stage of the course.  In Unit 7.4.3, section (d)iii, you learned about bijections from set A to set B.  If two sets A and B have a bijection between them, they are said to be equinumerous.  It turns out that the relation equinumerous is an equivalence relation on the collection of all subsets of the real numbers (in fact on any set of sets).  The equivalence classes (the cells) of this relation are called cardinalities.

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  • Unit 8: Combinatorics

    In this unit, you will learn to count.  That is, you will learn to classify the objects of a set in such a way that one of several principles applies.

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