...of a Set. In a) above, elements of Set C are 1,2,3 and 4. c) Order of a Set: are special binary relations. Suppose that P is a set and that ≤ is a relation on P, Then ≤ is a partial order if it is reflexive, antisymmetric and transitive. d) Null Set: is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Common notations for the empty set include "{}", " ", and " ". e) Finite Set: a finite set is a set that has a finite number of elements. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (non-negative integer),and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positive integers is infinite: f) Proper Subset: A proper subset is a grouping of numbers in which all the numbers for two quantities have the same numbers, but are not equal. g) Data: are values of qualitative or quantitative variables belonging to a set of items. h) Statistics: Statistics is a branch of mathematics that deals with the collection, organization and interpretation of data. i) Probability: is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen). The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater...
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...An Introduction to Real Analysis John K. Hunter 1 Department of Mathematics, University of California at Davis 1 The author was supported in part by the NSF. Thanks to Janko Gravner for a number of corrections and comments. Abstract. These are some notes on introductory real analysis. They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. They don’t include multi-variable calculus or contain any problem sets. Optional sections are starred. c John K. Hunter, 2014 Contents Chapter 1. Sets and Functions 1 1.1. Sets 1 1.2. Functions 5 1.3. Composition and inverses of functions 7 1.4. Indexed sets 8 1.5. Relations 11 1.6. Countable and uncountable sets 14 Chapter 2. Numbers 21 2.1. Integers 22 2.2. Rational numbers 23 2.3. Real numbers: algebraic properties 25 2.4. Real numbers: ordering properties 26 2.5. The supremum and infimum 27 2.6. Real numbers: completeness 29 2.7. Properties of the supremum and infimum 31 Chapter 3. Sequences 35 3.1. The absolute value 35 3.2. Sequences 36 3.3. Convergence and limits 39 3.4. Properties of limits 43 3.5. Monotone sequences 45 3.6. The lim sup and lim inf 48 3.7. Cauchy sequences ...
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...Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic[1] – do not vary smoothly in this way, but have distinct, separated values.[2] Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact definition of the term "discrete mathematics."[4] Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions. The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful...
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...Overview and examples from Finite Mathematics Using Microsoft Excel® Revathi Narasimhan Saint Peter's College An electronic supplement to Finite Mathematics and Its Applications, 6th Ed. , by Goldstein, Schneider, and Siegel, Prentice Hall, 1997 Introduction In any introductory mathematics course designed for non-mathematics majors, it is important for the student to understand and apply mathematical ideas in a variety of contexts. With the increased use of advanced software in all fields, it is also important for the student to effectively interact with the new technology. Our goal is to integrate these two objectives in a supplement for the text Finite Mathematics and Its Applications, by Goldstein, Schneider, and Siegel. The package consists of interactive tutorials and projects in an Excel workbook format. The software platform used is the Microsoft Excel 5.0 spreadsheet. It was chosen for the following reasons: • • • suited to applications encountered in a finite math course widespread use outside of academia ease of creating reports with a professional look Use of Excel 5.0 was put into effect in the author's sections of the Finite Mathematics II course in the Spring 1996 semester. It was expanded to cover the Finite Mathematics I course for the Fall semester of 1996. Using a combination of specially designed projects and tutorials, students are able to analyze data, draw conclusions, and present their analysis in a professional format. The mathematical...
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...Riemann was a phenomenal genius whose work was exceptionally deep, creative and rigorous; he made revolutionary contributions in many areas of pure mathematics, and also inspired the development of physics. He had poor physical health and died at an early age, yet is still considered to be among the most productive mathematicians ever. He was the master of complex analysis, which he connected to both topology and number theory. He applied topology to analysis, and analysis to number theory, making revolutionary contributions to all three fields. He took non-Euclidean geometry far beyond his predecessors. He introduced the Riemann integral which clarified analysis. Riemann's other masterpieces include differential geometry, tensor analysis, the theory of functions, and, especially, the theory of manifolds. He generalized the notions of distance and curvature and, therefore, described new possibilities for the geometry of space itself. Several important theorems and concepts are named after Riemann, e.g. the Riemann-Roch theorem, a key connection among topology, complex analysis and algebraic geometry. He was so prolific and original that some of his work went unnoticed (for example, Weierstrass became famous for showing a nowhere-differentiable continuous function; later it was found that Riemann had casually mentioned one in a lecture years earlier). Like his mathematical peers (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. His theory unifying electricity...
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...Set Theory, branch of mathematics, first given formal treatment by the German mathematician Georg Cantor in the 19th century. The set concept is one of the most basic in mathematics, even more primitive than the process of counting, and is found, explicitly or implicitly, in every area of pure and applied mathematics. Explicitly, the principles and terminology of sets are used to make mathematical statements more clear and precise and to clarify concepts such as the finite and the infinite. A set is an aggregate, class, or collection of objects, which are called the elements of the set. In symbols, aS means that the element a belongs to or is contained in the set S, or that the set S contains the element a. A set S is defined if, given any object a, one and only one of these statements holds: aS or aS (that is, a is not contained in S). A set is frequently designated by the symbol S = { }, with the braces including the elements of S either by writing all of them in explicitly or by giving a formula, rule, or statement that describes all of them. Thus, S1 = {2, 4}; S2 = {2, 4, 6, ..., 2n,...} = {all positive even integers}; S3 = {x | x2 - 6x + 11 ≥ 3}; S4 = {all living males named John}. In S3 and S4 it is implied that x is a number; S3 is read as the set of all xs such that x2 - 6x + 11 ≥ 3. If every element of a set R also belongs to a set S,R is a subset of S, and S is a superset of R; in symbols, RS, or SR. A set is both a subset and a superset of itself. If RS, but...
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...EFFICIENCY LEVEL IN SOLVING POLYNOMIAL EQUATIONS AND THEIR PERFORMANCE IN MATHEMATICS OF GRADE 9 STUDENTS A Thesis Presented to the Faculty of the Teacher Education Program Ramon Magsaysay Memorial Colleges General Santos City In Partial Fulfillment of the Requirement for the Degree Bachelor of Secondary Education Major in Mathematics Armando V. Delino Jr. October 2015 TABLE OF CONTENTS Contents Page Title Page i Table of contents ii CHAPTER I THE PROBLEM AND ITS SETTING 1 Introduction 1 Theoretical Framework Conceptual Framework Statement of the problem Hypothesis Significance of the study Scope of the study Definition of terms CHAPTER II REVIEW OF RELATED LITERATURE CHAPTER III METHODOLOGY Research Design Research Locale Sampling Technique Research Instrument Statistical Treatment CHAPTER 1 PROBLEM AND ITS SETTING Introduction In Mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials appear in a wide variety of areas of Mathematics and Science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the Sciences; they are used to define polynomial functions, which appear in...
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...Logical Operations and Truth Tables At first glance, it may not seem that the study of logic should be part of mathematics. For most of us, the word logic is associated with reasoning in a very nebulous way: "If my car is out of gas, then I cannot drive it to work." seems logical enough, while "If I am curious, then I am yellow." is clearly illogical. Yet our conclusions about what is or is not logical are most often unstructured and subjective. The purpose of logic is to enable the logician to construct valid arguments which satisfy the basic principle "If all of the premises are true, then the conclusion must be true." It turns out that in order to reliably and objectively construct valid arguments, the logical operations which one uses must be clearly defined and must obey a set of consistent properties. Thus logic is quite rightly treated as a mathematical subject. Up until now, you've probably considered mathematics as a set of rules for using numbers. The study of logic as a branch of mathematics will require you to think more abstractly then you are perhaps used to doing. For instance, in logic we use variables to represent propositions (or premises), in the same fashion that we use variables to represent numbers in algebra. But while an algebraic variable can have any number as its value, a logical variable can only have the value True or False. That is, True and False are the "numerical constants" of logic. And instead of the usual arithmetic operators (addition...
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...permutation In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). One might define an anagram of a word as a permutation of its letters. The study of permutations in this sense generally belongs to the field of combinatorics. The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×...×2×1, which number is called "n factorial" and written "n!". Permutations occur, in more or less prominent ways, in almost every domain of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science. In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself (i.e., a map S → S for which every element of S occurs exactly once as image value). This is related to the rearrangement of S in which each element s takes the place of the corresponding f(s). The collection of such permutations form a symmetric group. The key to its structure is the possibility...
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...with one column is called a column matrix Matrices of the same kindMatrix A and B are of the same kind if and only if A has as many rows as B and A has as many columns as B The tranpose of a matrix The n x m matrix A' is the transpose of the m x n matrix A if and only if The ith row of A = the ith column of A' for (i = 1,2,3,..n) So ai,j = aj,i' The transpose of A is denoted T(A) or AT 0-matrix When all the elements of a matrix A are 0, we call A a 0-matrix. We write shortly 0 for a 0-matrix. An identity matrix I An identity matrix I is a diagonal matrix with all diagonal element = 1. A scalar matrix S A scalar matrix S is a diagonal matrix with all diagonal elements alike. a1,1 = ai,i for (i = 1,2,3,..n) Matrix in mathematics, a system of elements aij (numbers, functions, or other quantities for which there are defined algebraic operations) arranged in the form of a...
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...Mathematics HL First examinations 2008 b DIPLOMA PROGRAMME MATHEMATICS HL First examinations 2008 International Baccalaureate Organization Buenos Aires Cardiff Geneva New York Singapore Diploma Programme Mathematics HL First published in September 2006 International Baccalaureate Organization Peterson House, Malthouse Avenue, Cardiff Gate Cardiff, Wales GB CF23 8GL United Kingdom Phone: + 44 29 2054 7777 Fax: + 44 29 2054 7778 Web site: www.ibo.org c International Baccalaureate Organization 2006 The International Baccalaureate Organization (IBO) was established in 1968 and is a non-profit, international educational foundation registered in Switzerland. The IBO is grateful for permission to reproduce and/or translate any copyright material used in this publication. Acknowledgments are included, where appropriate, and, if notified, the IBO will be pleased to rectify any errors or omissions at the earliest opportunity. IBO merchandise and publications in its official and working languages can be purchased through the IB store at http://store.ibo.org. General ordering queries should be directed to the sales and marketing department in Cardiff. Phone: +44 29 2054 7746 Fax: +44 29 2054 7779 E-mail: sales@ibo.org Printed in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire. 5007 CONTENTS INTRODUCTION 1 NATURE OF THE SUBJECT 3 AIMS 6 OBJECTIVES 7 SYLLABUS OUTLINE 8 SYLLABUS DETAILS 9 ASSESSMENT OUTLINE 53 ASSESSMENT DETAILS ...
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...Kelley England ENG-200/Dr. Roemer 12/14/05 “Cargoes” By: John Masefield In the poem “Cargoes,” the author, John Masefield, writes about nostalgia and progress. The gritty realities of modern life are set against the golden age of the “Stately Spanish galleon” and the even more distant glamour of the “Quinquereme of Ninevah.” The author evokes the senses using multiple kinds of imagery in each stanza of the poem. The poem encompasses the past and the future of the changing ships and cargoes throughout various periods of history. John Masefield was born in 1878 in Ledbury, England. He married his wife, Constance, at the age of 23 and had two children, a boy and a girl. He suffered tragedies early in his life such as his mother’s death when he was 6, both of his grandparents’ deaths at the age of 7, and his father having a mental breakdown five years later and then dying when John was 12 years old. His Aunt and Uncle took on the responsibility of raising him and at the age of 13, his Aunt sent him to the sea-cadet ship the HMS Conway to train for a life at sea. It was aboard this ship that he developed a love for story telling. Sea life did not suit John and on his second voyage he deserted his ship in New York City and began to travel the countryside, taking whatever odd jobs he could find, often sleeping outdoors and eating very little. After 3 years, he was ready to return to England. John became a very big admirer of William Butler Yeats and after many letters...
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...Babylonians, and the people of the Indus Valley but the creators were Pythagoras, Euclid, Archimedes, and Thales. Pythagoras was the first pure mathematician although we know little about his mathematical achievements. He was also, a greek philosopher and created a movement called Pythagoreanism. Euclid is sometimes called Euclid of Alexandria. He is also called the “Father of Geometry” and his elements were one of the most influential works in the history of mathematics, which served as a textbook used for teaching mathematics (especially Geometry) from when it was published till the late 19th century to early 20th century. In the Elements he included the principles of what is now called Euclidean Geometry. Euclidean Geometry is a mathematical system and consists of in a small set of appealing postulates that are accepted as true. In fact, Euclid was able to come up with a great portion of plane geometry from five postulates. These postulates include: A straight line segment can be drawn joining any two points, to extend a finite straight line continuously in a straight line, given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center, all right angles are congruent, and if two lines are drawn which intersect a...
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...LOGICAL INFERENCE & PROOFs Debdeep Mukhopadhyay Dept of CSE, IIT Madras Defn • A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of a theorem. Nature & Importance of Proofs • In mathematics, a proof is: – a correct (well-reasoned, logically valid) and complete (clear, detailed) argument that rigorously & undeniably establishes the truth of a mathematical statement. • Why must the argument be correct & complete? – Correctness prevents us from fooling ourselves. – Completeness allows anyone to verify the result. • In this course (& throughout mathematics), a very high standard for correctness and completeness of proofs is demanded!! Overview • Methods of mathematical argument (i.e., proof methods) can be formalized in terms of rules of logical inference. • Mathematical proofs can themselves be represented formally as discrete structures. • We will review both correct & fallacious inference rules, & several proof methods. Applications of Proofs • An exercise in clear communication of logical arguments in any area of study. • The fundamental activity of mathematics is the discovery and elucidation, through proofs, of interesting new theorems. • Theorem-proving has applications in program verification, computer security, automated reasoning systems, etc. • Proving a theorem allows us to rely upon on its correctness even in the most critical scenarios. Proof Terminology • Theorem ...
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... mathematical and linear algebra formulas, calculations, graphs, diagrams, pictures, etc., will be clearly shown as to further understand the applicability of linear algebra in economics. Calculations and mathematical examples used in economics will be provided in the context of this paper for better understanding. Also, terms and notations used will be explained, derivation and origin of mathematical results will be shown. Definitions: Economics is a branch of knowledge concerned with the production, distributions, and consumption of goods and services. Linear algebra is a branch of mathematics with the properties of finite dimensional vector spaces and linear mapping between the spaces. The equations are represented using matrices and vectors and consist of several unknowns. Econometrics is branch of economics that aims to give the definitions of application of mathematics and statistical methods to economic data. Game theory is a method of studying strategic decision-making, that is, the study of mathematical models of conflict and cooperation between decision makers (Levine). Input-output Analysis Application: The economy is a complicated network of sectors that interact with each...
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