The term recursion refers to a method of defining sequences of numbers, functions, and other objects. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. The axiomatic approach to will be postponed until the unit on recursion and mathematical induction. Some of these relations are functions from to. There are several reasons for including puzzles. Finally, you will study several methods of proof including proof by contradiction, proof by complete enumeration, etc.
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Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: In this unit, you will prove propositions about an infinite set of positive integers.
The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. A great puzzle is like a laboratory for proving propositions. Relations that satisfy R, S, and T are called equivalence relations, and those satisfying R, A, and T are called partial orderings.
In this unit, you will learn to count.
In this unit, you will ka111 by considering various puzzles, including Ken-Ken and Sudoku. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions.
The distinction between a fraction and a rational number will also be discussed. Some aspects of the solutions motivate ideas you will encounter later in the course.
Mathematical induction is a technique used to formulate all such proofs. Each topic in this course is standard except the first one, puzzles.
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. Introduction to Mathematical Reasoning. Introduction to Number Theory This unit is na111 concerned with the set of natural numbers. 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 ma11.
Calculus I – MA 111
In Unit 7, you learned about bijections from set to set. That means that strictly between any two real numbers, you can always find a rational number. If two sets and have a bijection between them, they are said to be equinumerous. Another reason for including puzzles is to foster creativity. The main purpose of this course is to na111 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.
Completing this unit should take you approximately 31 hours.
This unit is primarily concerned with the set of natural numbers. In fact, there are many different size infinite sets. Completing this unit should take you approximately 10 hours. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets.
Sometimes the conjectures turn out to be provable, but often they do ma111, and the mathematician goes back to playing.
This fundamental idea enables you mq111 completely understand the algorithms we learned ma1111 elementary school for addition, subtraction, multiplication, and division of multi-digit integers. You will learn the importance of tenacity in approaching mathematical problems including puzzles and brain teasers. Logic In this unit, you will begin by considering various puzzles, including Ken-Ken and Sudoku.
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.
This multi-step process is perfectly mirrored in solving the KenKen problems this course presents. Mathematical Induction In this unit, you will prove propositions m1a11 an infinite set of positive integers.
This unit will help you understand the multiplicative and additive structure of. You will see examples of self-referencing sentences which are not propositions.