This an animated slides for students. Introduce basis concept of proofs to students. Direct proofs. Please search for slides Proof methods-teachers. If you want to teach using these slides.
These slides about functions between two sets. Topics included are
1- The definition of a function as relations between two sets
2- Examples of infinite functions
3- Relations which are not functions
4- Domain and range of real functions
5- Identical functions
Videos explaining these slides are available in the following links
1- The definition of a function as relations between two sets
https://youtu.be/Vi3N2vLySd0
2- Examples of infinite functions
https://youtu.be/Fex82-Ml55c
3- Relations which are not functions
https://youtu.be/abhbALKcHn8
4- Domain and range of real functions
https://youtu.be/82LJ5MXAKRQ
5- Identical functions
https://youtu.be/ZOIt5JxoBxo
The reference book for these slides is
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/i...
These slides cover the relation between two sets, domain and range of a relation, the inverse of a relation, and composition of relations.
Videos explaining these slides are available
Part1: Definition of relations https://youtu.be/MR0ALeKvaSc
Part2: Domain and range of a relation https://youtu.be/kBlUWTDn-Z4
Part3: The inverse of a relation https://youtu.be/Uv5KFusvvsY
Part4: Composition of relations https://youtu.be/IvVIjJqqPH0
Reference for these slides are
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/i...
1. The document discusses functions and their notation. It defines functions as mappings from inputs to outputs and provides examples of function notation.
2. It also discusses different types of relations between inputs and outputs such as one-to-one, one-to-many, many-to-one, and many-to-many relations.
3. The document explains concepts related to functions such as composite functions, inverse functions, and how to determine composite and inverse functions through examples.
The document contains 14 mathematics word problems involving arithmetic progressions. The problems cover finding common differences, sums of terms, individual terms, and relating terms to each other. They range from 3 to 4 marks and include SPM past year questions from 2003 to 2006.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
X std mathematics - Relations and functions (Ex 1.4), Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, representation of functions, set or ordered pair, table form, arrow diagram, graph, vertical line test, types of function, one -one function, many- one function, onto function, surjection, into function, horizontal line test, special cases of function,
The document discusses finite fields and related algebraic concepts. It begins by defining groups, rings, and fields. It then focuses on finite fields, particularly GF(p) fields consisting of integers modulo a prime p. It discusses finding multiplicative inverses in such fields using the extended Euclidean algorithm. As an example, it finds the inverse of 550 modulo 1759.
These slides about functions between two sets. Topics included are
1- The definition of a function as relations between two sets
2- Examples of infinite functions
3- Relations which are not functions
4- Domain and range of real functions
5- Identical functions
Videos explaining these slides are available in the following links
1- The definition of a function as relations between two sets
https://youtu.be/Vi3N2vLySd0
2- Examples of infinite functions
https://youtu.be/Fex82-Ml55c
3- Relations which are not functions
https://youtu.be/abhbALKcHn8
4- Domain and range of real functions
https://youtu.be/82LJ5MXAKRQ
5- Identical functions
https://youtu.be/ZOIt5JxoBxo
The reference book for these slides is
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/i...
These slides cover the relation between two sets, domain and range of a relation, the inverse of a relation, and composition of relations.
Videos explaining these slides are available
Part1: Definition of relations https://youtu.be/MR0ALeKvaSc
Part2: Domain and range of a relation https://youtu.be/kBlUWTDn-Z4
Part3: The inverse of a relation https://youtu.be/Uv5KFusvvsY
Part4: Composition of relations https://youtu.be/IvVIjJqqPH0
Reference for these slides are
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/i...
1. The document discusses functions and their notation. It defines functions as mappings from inputs to outputs and provides examples of function notation.
2. It also discusses different types of relations between inputs and outputs such as one-to-one, one-to-many, many-to-one, and many-to-many relations.
3. The document explains concepts related to functions such as composite functions, inverse functions, and how to determine composite and inverse functions through examples.
The document contains 14 mathematics word problems involving arithmetic progressions. The problems cover finding common differences, sums of terms, individual terms, and relating terms to each other. They range from 3 to 4 marks and include SPM past year questions from 2003 to 2006.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
X std mathematics - Relations and functions (Ex 1.4), Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, representation of functions, set or ordered pair, table form, arrow diagram, graph, vertical line test, types of function, one -one function, many- one function, onto function, surjection, into function, horizontal line test, special cases of function,
The document discusses finite fields and related algebraic concepts. It begins by defining groups, rings, and fields. It then focuses on finite fields, particularly GF(p) fields consisting of integers modulo a prime p. It discusses finding multiplicative inverses in such fields using the extended Euclidean algorithm. As an example, it finds the inverse of 550 modulo 1759.
Basic galois field arithmatics required for error control codesMadhumita Tamhane
ย
Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
This document discusses the chain rule for finding derivatives. It explains that the chain rule is needed when taking the derivative of a composition of functions, where an "inside function" is plugged into an "outside function". The chain rule formula is given as the derivative of the outside function multiplied by the derivative of the inside function. Several examples are worked through, applying the chain rule when the power rule alone cannot be used, such as when the base of an exponent is a function rather than a variable. The document also notes that problems may require using multiple derivative rules, like the product rule and chain rule, to fully solve them.
This document provides an overview of key concepts in multivariable calculus covered in MAT 10B at Williams College in Fall 2011, including:
1) Definitions and computations of double integrals, including over rectangles and more general regions, and Fubini's theorem on changing the order of integration.
2) Triple integrals as a generalization of double integrals to functions of three variables over boxes in R3.
3) Techniques for changing the order of integration in iterated integrals when one order may be more convenient than another.
This document provides an overview of functions and relations. It begins by defining the learning objectives and outcomes for understanding functions. It then discusses representing relations using arrow diagrams, ordered pairs, and graphs. It introduces the concepts of domain, codomain, object, image, and range for relations. Different types of relations like one-to-one, many-to-one, one-to-many, and many-to-many are classified. Functions are introduced as a special type of relation where each element in the domain maps to only one element in the codomain. Notation for expressing functions is explained along with determining the domain, object, image, and range of functions. Examples are provided to illustrate these concepts.
Cryptography and data security involves number theory concepts like groups, rings, fields, and modular arithmetic. Some key ideas discussed include:
1) The integers under addition form a cyclic group, and the theorem that for any finite group G and element a in G, a raised to the order of G is the identity element.
2) Modular arithmetic defines equivalence classes for integers modulo n, and the set of residues Zn forms an abelian group under addition.
3) The multiplicative integers modulo n, Zn*, form a group whose size is given by Euler's totient function ฯ(n). For prime p, ฯ(p) = p - 1.
Mat221 5.6 definite integral substitutions and the area between two curvesGlenSchlee
ย
This document contains examples of evaluating definite integrals using substitution and finding the area between curves. It includes 8 examples of evaluating definite integrals using techniques like u-substitution, integration by parts, and trigonometric substitutions. It also contains 3 examples of finding the area of regions bounded by graphs by setting up and evaluating definite integrals with respect to x or y.
This document discusses inverses of functions. It provides examples of finding the inverse of various functions by switching the x and y coordinates, solving for y, and determining if the inverse is a function. Key points made are: to find the inverse change the coordinate pair; a function and its inverse are reflections over y=x; when composing a function with its inverse, you get back the original function. Examples are worked through and conclusions are drawn about the domains and ranges of inverses.
Star, a wild bird, learned to count up to 8 on her own and discovered that numbers can be represented in different ways like 4+4 or 2+2+2+2, showing she was thinking about numbers consciously. She could also recognize number names and remember their sounds. Star showed unusual intelligence for a wild bird in her self-motivated pursuit of numerical science.
This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.
The document discusses congruences and the Chinese Remainder Theorem. It begins by introducing congruences and some basic properties, such as if a โก b (mod m) and c โก d (mod m), then a + c โก b + d (mod m). It then discusses the Euler phi function and Euler's Theorem. Finally, it introduces and proves the Chinese Remainder Theorem, which states that a system of congruences with pairwise relatively prime moduli has a unique solution modulo the product of the moduli.
This document provides an introduction to partial fractions. It defines key terms like polynomials, rational functions, and proper and improper fractions. It then outlines the three main cases for splitting a fraction into partial fractions: (1) a linear factor (ax+b), (2) a repeated linear factor, and (3) a quadratic factor (ax^2+bx+c). For each case, it provides an example of how to write the fraction as a sum of partial fractions. It concludes by emphasizing two important checks: (1) the fraction must be proper, and (2) the denominator must be completely factorized before attempting to write it as partial fractions.
The document discusses applying the distributive property to simplify algebraic expressions. It provides examples of distributing positive and negative coefficients over terms with variables and constants. Key parts of expressions like terms, coefficients, and constant terms are defined. Examples show identifying these parts and simplifying expressions using the distributive property and combining like terms. Practice problems at the end ask the reader to simplify expressions.
The document discusses relations and functions. It defines relations as subsets of Cartesian products of sets and describes how to classify relations as reflexive, symmetric, transitive, or an equivalence relation. It also defines functions, including their domain, codomain, and range. It describes how to classify functions as injective, surjective, or bijective. Examples are provided to illustrate these concepts of relations and functions.
This document contains notes on probability theory from a course. It begins with definitions of measures, ฯ-algebras, Borel ฯ-algebras, and related concepts. It then proves some key properties, including that the Borel ฯ-algebra on the real line can be generated by open intervals with rational endpoints. The document also contains proofs showing when two measures are equal and the Monotone Class Theorem for sets.
Introductory Mathematical Analysis for Business Economics International 13th ...Leblancer
ย
Full download : http://alibabadownload.com/product/introductory-mathematical-analysis-for-business-economics-international-13th-edition-haeussler-test-bank/ Introductory Mathematical Analysis for Business Economics International 13th Edition Haeussler Test Bank
1) The document is a mathematics worksheet for class 11 that contains 9 multiple choice questions about relations and functions.
2) The questions cover topics like determining the domain and range of various functions, identifying which sets of ordered pairs represent functions, and properties of relations such as the inverse of a relation.
3) The worksheet is intended to test students' understanding of key concepts in relations and functions through multiple choice problems.
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
Cryptography and Data Security often relies on number theory concepts. This document reviews several key number theory topics used in cryptography, including: 1) integers modulo n and Euler's totient function; 2) Euler's theorem and Fermat's theorem; 3) the greatest common divisor and Euclid's algorithm; and 4) polynomials and finite fields. Finite fields play an important role in cryptography by allowing the representation of data as field elements.
Understanding the "Chain Rule" for Derivatives by Deriving Your Own VersionJames Smith
ย
Because the Chain Rule can confuse students as much as it helps them solve real problems, we put ourselves in the shoes of the mathematicians who derived it, so that students may understand the motivation for the rule; its limitations; and why textbooks present it in its customary form. We begin by finding the derivative of sin2x without using the Chain Rule. That exercise, having shown that even a comparatively simple compound function can be bothersome to differentiate using the definition of the derivative as a limit, provides the motivation for developing our own formula for the derivative of the general compound function g[f(x)]. In the course of that development, we see why the function f must be continuous at any value of x to which the formula is applied. We finish by comparing our formula to that which is commonly given.
A function f associates a unique output with each input. The input is denoted as x and the output is denoted as f(x). The domain of a function f is the set of all allowable input values, while the range is the set of all output values. An inverse function f-1 can be written if f is one-to-one and onto. The composition of two functions f and g, denoted as fog and gof, are defined as fog=f(g(x)) and gof=g(f(x)).
1. The document discusses functions and their properties. It defines a function as a special relation that maps each element in a set X to only one element in a set Y.
2. It provides examples of functions defined using function notation and discusses how to determine if a relation or graph represents a function using the vertical line test.
3. The document contains examples of finding domain, codomain, range, images and objects of various functions. It also includes examples of sketching graphs of functions and solving problems involving functions.
Basic galois field arithmatics required for error control codesMadhumita Tamhane
ย
Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
This document discusses the chain rule for finding derivatives. It explains that the chain rule is needed when taking the derivative of a composition of functions, where an "inside function" is plugged into an "outside function". The chain rule formula is given as the derivative of the outside function multiplied by the derivative of the inside function. Several examples are worked through, applying the chain rule when the power rule alone cannot be used, such as when the base of an exponent is a function rather than a variable. The document also notes that problems may require using multiple derivative rules, like the product rule and chain rule, to fully solve them.
This document provides an overview of key concepts in multivariable calculus covered in MAT 10B at Williams College in Fall 2011, including:
1) Definitions and computations of double integrals, including over rectangles and more general regions, and Fubini's theorem on changing the order of integration.
2) Triple integrals as a generalization of double integrals to functions of three variables over boxes in R3.
3) Techniques for changing the order of integration in iterated integrals when one order may be more convenient than another.
This document provides an overview of functions and relations. It begins by defining the learning objectives and outcomes for understanding functions. It then discusses representing relations using arrow diagrams, ordered pairs, and graphs. It introduces the concepts of domain, codomain, object, image, and range for relations. Different types of relations like one-to-one, many-to-one, one-to-many, and many-to-many are classified. Functions are introduced as a special type of relation where each element in the domain maps to only one element in the codomain. Notation for expressing functions is explained along with determining the domain, object, image, and range of functions. Examples are provided to illustrate these concepts.
Cryptography and data security involves number theory concepts like groups, rings, fields, and modular arithmetic. Some key ideas discussed include:
1) The integers under addition form a cyclic group, and the theorem that for any finite group G and element a in G, a raised to the order of G is the identity element.
2) Modular arithmetic defines equivalence classes for integers modulo n, and the set of residues Zn forms an abelian group under addition.
3) The multiplicative integers modulo n, Zn*, form a group whose size is given by Euler's totient function ฯ(n). For prime p, ฯ(p) = p - 1.
Mat221 5.6 definite integral substitutions and the area between two curvesGlenSchlee
ย
This document contains examples of evaluating definite integrals using substitution and finding the area between curves. It includes 8 examples of evaluating definite integrals using techniques like u-substitution, integration by parts, and trigonometric substitutions. It also contains 3 examples of finding the area of regions bounded by graphs by setting up and evaluating definite integrals with respect to x or y.
This document discusses inverses of functions. It provides examples of finding the inverse of various functions by switching the x and y coordinates, solving for y, and determining if the inverse is a function. Key points made are: to find the inverse change the coordinate pair; a function and its inverse are reflections over y=x; when composing a function with its inverse, you get back the original function. Examples are worked through and conclusions are drawn about the domains and ranges of inverses.
Star, a wild bird, learned to count up to 8 on her own and discovered that numbers can be represented in different ways like 4+4 or 2+2+2+2, showing she was thinking about numbers consciously. She could also recognize number names and remember their sounds. Star showed unusual intelligence for a wild bird in her self-motivated pursuit of numerical science.
This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.
The document discusses congruences and the Chinese Remainder Theorem. It begins by introducing congruences and some basic properties, such as if a โก b (mod m) and c โก d (mod m), then a + c โก b + d (mod m). It then discusses the Euler phi function and Euler's Theorem. Finally, it introduces and proves the Chinese Remainder Theorem, which states that a system of congruences with pairwise relatively prime moduli has a unique solution modulo the product of the moduli.
This document provides an introduction to partial fractions. It defines key terms like polynomials, rational functions, and proper and improper fractions. It then outlines the three main cases for splitting a fraction into partial fractions: (1) a linear factor (ax+b), (2) a repeated linear factor, and (3) a quadratic factor (ax^2+bx+c). For each case, it provides an example of how to write the fraction as a sum of partial fractions. It concludes by emphasizing two important checks: (1) the fraction must be proper, and (2) the denominator must be completely factorized before attempting to write it as partial fractions.
The document discusses applying the distributive property to simplify algebraic expressions. It provides examples of distributing positive and negative coefficients over terms with variables and constants. Key parts of expressions like terms, coefficients, and constant terms are defined. Examples show identifying these parts and simplifying expressions using the distributive property and combining like terms. Practice problems at the end ask the reader to simplify expressions.
The document discusses relations and functions. It defines relations as subsets of Cartesian products of sets and describes how to classify relations as reflexive, symmetric, transitive, or an equivalence relation. It also defines functions, including their domain, codomain, and range. It describes how to classify functions as injective, surjective, or bijective. Examples are provided to illustrate these concepts of relations and functions.
This document contains notes on probability theory from a course. It begins with definitions of measures, ฯ-algebras, Borel ฯ-algebras, and related concepts. It then proves some key properties, including that the Borel ฯ-algebra on the real line can be generated by open intervals with rational endpoints. The document also contains proofs showing when two measures are equal and the Monotone Class Theorem for sets.
Introductory Mathematical Analysis for Business Economics International 13th ...Leblancer
ย
Full download : http://alibabadownload.com/product/introductory-mathematical-analysis-for-business-economics-international-13th-edition-haeussler-test-bank/ Introductory Mathematical Analysis for Business Economics International 13th Edition Haeussler Test Bank
1) The document is a mathematics worksheet for class 11 that contains 9 multiple choice questions about relations and functions.
2) The questions cover topics like determining the domain and range of various functions, identifying which sets of ordered pairs represent functions, and properties of relations such as the inverse of a relation.
3) The worksheet is intended to test students' understanding of key concepts in relations and functions through multiple choice problems.
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
Cryptography and Data Security often relies on number theory concepts. This document reviews several key number theory topics used in cryptography, including: 1) integers modulo n and Euler's totient function; 2) Euler's theorem and Fermat's theorem; 3) the greatest common divisor and Euclid's algorithm; and 4) polynomials and finite fields. Finite fields play an important role in cryptography by allowing the representation of data as field elements.
Understanding the "Chain Rule" for Derivatives by Deriving Your Own VersionJames Smith
ย
Because the Chain Rule can confuse students as much as it helps them solve real problems, we put ourselves in the shoes of the mathematicians who derived it, so that students may understand the motivation for the rule; its limitations; and why textbooks present it in its customary form. We begin by finding the derivative of sin2x without using the Chain Rule. That exercise, having shown that even a comparatively simple compound function can be bothersome to differentiate using the definition of the derivative as a limit, provides the motivation for developing our own formula for the derivative of the general compound function g[f(x)]. In the course of that development, we see why the function f must be continuous at any value of x to which the formula is applied. We finish by comparing our formula to that which is commonly given.
A function f associates a unique output with each input. The input is denoted as x and the output is denoted as f(x). The domain of a function f is the set of all allowable input values, while the range is the set of all output values. An inverse function f-1 can be written if f is one-to-one and onto. The composition of two functions f and g, denoted as fog and gof, are defined as fog=f(g(x)) and gof=g(f(x)).
1. The document discusses functions and their properties. It defines a function as a special relation that maps each element in a set X to only one element in a set Y.
2. It provides examples of functions defined using function notation and discusses how to determine if a relation or graph represents a function using the vertical line test.
3. The document contains examples of finding domain, codomain, range, images and objects of various functions. It also includes examples of sketching graphs of functions and solving problems involving functions.
The document discusses important concepts related to relations and functions. It defines what a relation is and different types of relations such as reflexive, symmetric, transitive, and equivalence relations. It also defines different types of functions including one-to-one, onto, bijective, and inverse functions. It provides examples of binary operations and discusses their properties like commutativity, associativity, and identity elements. It concludes with short answer and very short answer type questions related to these concepts.
1. The document discusses various mathematical functions including absolute value, floor/ceiling functions, factorials, modular arithmetic, exponential functions, logarithmic functions, and polynomials.
2. It defines key properties of functions such as one-to-one, onto, bijective, and inverse functions. It also covers function composition and important sequences like geometric and arithmetic progressions.
3. The document provides formulas for summing sequences and common summations like the sum of the first n natural numbers and the sum of the first n odd integers.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
This chapter discusses functions and graphs. It introduces different types of functions including constant, special, rational, and absolute value functions. It covers combining functions using addition, subtraction, multiplication, and composition. Inverse functions and their properties are explained. Graphing functions in a rectangular coordinate system and finding intercepts is demonstrated. The chapter also discusses symmetry of graphs about the x-axis, y-axis, origin, and line y = x.
The document discusses functions and graphs in chapter 2 of an introductory mathematical analysis textbook. It introduces key concepts such as functions, domains, ranges, combinations of functions, inverse functions, and graphs in rectangular coordinates. It provides examples of determining equality of functions, finding function values, combining functions, and finding inverses. It also discusses special functions, graphs, symmetry, and intercepts. The chapter aims to define functions and domains, introduce different types of functions and their operations, and familiarize students with graphing equations and basic function shapes.
This chapter discusses functions and graphs. It introduces different types of functions including constant, special, rational, and absolute value functions. It covers combining functions using addition, subtraction, multiplication, and composition. Inverse functions and their properties are explained. Graphing functions in a rectangular coordinate system and finding intercepts is demonstrated. The chapter also discusses symmetry of graphs about the x-axis, y-axis, origin, and line y = x.
This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
This document discusses functions and relations. It defines a relation as a correspondence between two sets, with an element x in one set corresponding to an element y in the other set. A function is defined as a special type of relation where each element x in the domain corresponds to exactly one element y in the range. Several examples are provided to illustrate determining if a relation represents a function, and specifying the domain and range if it does. The document also covers evaluating functions for given values and performing operations on functions.
This document provides information about functions from Additional Mathematics Module Form 4. It defines relations and different ways to represent them using arrow diagrams, ordered pairs, and graphs. It explains the concepts of domain, codomain, range, objects and images. Functions are defined as a special type of relation where each object has only one image. Function notation and evaluation are demonstrated through examples. Composite functions are introduced as the combined effect of two functions gf(x). Examples are provided to show how to determine one function given information about the composite function.
This document provides information about functions from Additional Mathematics Module Form 4. It defines relations and ways to represent them using arrow diagrams, ordered pairs, and graphs. It explains the concepts of domain, codomain, range, objects and images. Functions are defined as a special type of relation where each object has only one image. Function notation and evaluation are demonstrated through examples. Composite functions are introduced as the combined effect of two functions gf(x). Examples are provided to illustrate determining composite functions and using given information to find one of the functions. Exercises provide practice applying the concepts.
This document provides an introduction to functions and their properties. It defines what a function is as a mapping from a domain set to a codomain set, and introduces related concepts like domain, codomain, range, and the notation for functions. It then gives examples of specifying functions explicitly and with formulas. The document discusses properties of functions like injectivity, surjectivity, and being increasing or decreasing. It provides examples of determining if a function has these properties. The document concludes by introducing the concept of the inverse function for bijective functions.
This document provides definitions and examples of various types of numbers and functions. It discusses:
- Number sets including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
- Types of intervals such as closed, open, and semi-open/semi-closed.
- Definitions of a function, including domain, co-domain, and range. Methods of representing functions include mapping, algebraic, and ordered pairs.
- Classification of functions as algebraic vs. transcendental, even vs. odd, explicit vs. implicit, continuous vs. discontinuous, and increasing vs. decreasing.
- Properties of even and odd functions are also discussed.
The document introduces functions and some of their properties. It defines a function as a relation where each element in the domain is mapped to exactly one element in the codomain. It discusses one-to-one, onto, and bijective functions. It also covers composition, associativity, identity functions, and the pigeonhole principle as it relates to functions between finite sets.
1. The document contains a chapter on relations and functions with 36 short questions covering topics like relations, functions, one-to-one, onto, inverse functions, binary operations, and equivalence relations.
2. The questions range from very short answer type questions to short answer type questions requiring explanations and proofs.
3. The questions test concepts like checking if a relation is reflexive, determining if an element is part of a relation, evaluating composite functions, checking if a function is bijective or invertible, finding inverse functions, and determining identity elements and invertible elements for binary operations.
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
The chapter Lifelines of National Economy in Class 10 Geography focuses on the various modes of transportation and communication that play a vital role in the economic development of a country. These lifelines are crucial for the movement of goods, services, and people, thereby connecting different regions and promoting economic activities.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
ย
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analyticsโ feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
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Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
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(๐๐๐ ๐๐๐) (๐๐๐ฌ๐ฌ๐จ๐ง ๐)-๐๐ซ๐๐ฅ๐ข๐ฆ๐ฌ
๐๐ข๐ฌ๐๐ฎ๐ฌ๐ฌ ๐ญ๐ก๐ ๐๐๐ ๐๐ฎ๐ซ๐ซ๐ข๐๐ฎ๐ฅ๐ฎ๐ฆ ๐ข๐ง ๐ญ๐ก๐ ๐๐ก๐ข๐ฅ๐ข๐ฉ๐ฉ๐ข๐ง๐๐ฌ:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
๐๐ฑ๐ฉ๐ฅ๐๐ข๐ง ๐ญ๐ก๐ ๐๐๐ญ๐ฎ๐ซ๐ ๐๐ง๐ ๐๐๐จ๐ฉ๐ ๐จ๐ ๐๐ง ๐๐ง๐ญ๐ซ๐๐ฉ๐ซ๐๐ง๐๐ฎ๐ซ:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
1. 4.1 Functions as Relations
SQU-Math2350
Dr. Yassir Dinar
Sultan Qaboos University
Department Of Mathematics
Math2350: Foundation of Mathematics
Fall 2019
SQU-Math2350 4.1 Functions as Relations Fall 2019 1 / 12
2. Functions as Relations
Definition 1.1
A function (or mapping ) from A to B is a relation from A to B such
that
(i) the domain of f is A and
(ii) if (x, y) โ f and (x, y) โ f, then y = z.
f : A โ B readsโf is a function from A to Bโ or โf maps A to Bโ.
The set B is called the codomain of f and if A = B, we say f is a
function on A.
When (x, y) โ f, we write y = f(x). We read it as y is the image of f
at x (or value of f at x) and that x is a pre-image of y..
Functions whose domains and codomains are subsets of R are referred
to as real functions
SQU-Math2350 4.1 Functions as Relations Fall 2019 2 / 12
3. Relations: Examples
Example 1.2
Let A = {1, 2, 3} and B = {4, 5, 6} which of the following relations is a function
from A to B:
1 P = {(1, 4), (2, 5), (3, 6), (2, 6)}.
2 S = {(1, 5), (2, 5), (3, 4)}.
3 T = {(1, 4), (3, 6)}.
Solution.
SQU-Math2350 4.1 Functions as Relations Fall 2019 3 / 12
4. Functions: Examples
Example 1.3
Show that the relations g = {(a, b) : b + 2a โ 7 = 0} from N to Z is a function.
Solution.
(i) Let a โ N. Then b = โ2a + 7 โ Z and (a, b) โ g. Hence,
Dom(g) = N.
(ii) Assume (a, n) โ g and (a, m) โ g. Then by definition of g,
m = โ2a + 7 = n.
Frome (i) and (ii) we conclude that g is a function from N to Z.
Note that since g is a function we can write b = g(a) = โ2a + 7.
Example 1.4
Define F = {(x, y) : x2
+ 4y2
= 16}. Show that F is a function from [0, 4] to
[0, โ).
Since it is a function, write y =
q
16โx2
4 . Find 3 different codomains!!!
SQU-Math2350 4.1 Functions as Relations Fall 2019 4 / 12
5. Functions: Non Examples
Example 1.5
Define F = {(x, y) : x2
+ 4y2
= 16}. Show that F is not a function when
1 F is a relation from R to R.
2 F is a relation from [โ4, 4] to R.
3 F is a relation from [0, 4] to [0, 1].
Solution.
1 Note that 10 โ R but 10 /
โ Dom(F) since 102 + 4y2 โฅ 100, โy โ R.
Hence F is not a function.
2 Dom(F) = [โ4, 4] Prove it!. For (x, y) and (x, z) in F we have
4y2 = 4z2. Hence y = ยฑz. So F is not a function. Counterexample
both (2,
โ
3) and (2, โ
โ
3) are in F.
3 Dom(F) 6= [0, 4]. Counterexample 2 /
โ Dom(F) since otherwise
(2,
โ
3) โ F but
โ
3 /
โ codomain(F).
SQU-Math2350 4.1 Functions as Relations Fall 2019 5 / 12
6. Domain and Range
Example 1.6
Assume that the domain of each of the following functions is the larges possible
subset of R, find the domain and range.
1 f(x) = x2
+5x+6
x+2 .
2 f(x) =
โ
x โ 3.
Solution.
1 Dom(f) = R โ {โ2}. Let b โ Rng(f). Then there exists
a โ R โ {โ2} such that f(a) = b.This implies that b = a + 3. But
a 6= โ2. Hence, b 6= โ1. Therefore, Rng(f) = R โ {1}.
2
SQU-Math2350 4.1 Functions as Relations Fall 2019 6 / 12
7. Equal functions
Definition 1.7
Two functions f and g are equal if they are equal as sets.
Theorem 1.8
Two function f and g are equal if and only if
(i) Dom(f) = Dom(g) and
(ii) for all x โ Dom(f), f(x) = g(x).
Prove it!
Example 1.9
The function f(x) = x2
+5x+6
x+2 and g(x) = x + 3 are not equal as real values
functions since Dom(f) = R โ {โ2} while Dom(g) = R. If we redefine both of
them with domain [0, โ), then they are equal.
SQU-Math2350 4.1 Functions as Relations Fall 2019 7 / 12
8. Section 4.2 Constructions of Functions
Let f : A โ B and g : B โ C. Then fโ1 = {(a, b) : (b, a) โ f} ,
g โฆ f = {(x, y) : (โz โ B), (x, z) โ f and (z, y) โ g}.
Example 2.1
Let f and g be real valued functions given by f(x) = 2x + 1 and g(x) = x2
.
1 Find fโ1
and gโ1
and determine if they are functions.
2 Find g โฆ f and f โฆ g and determine if they are functions.
Solution.
Inverse of function is not necessary a function.
SQU-Math2350 4.2 Constructions of Functions Fall 2019 8 / 12
9. Composition of functions
Theorem 2.2
Let f : A โ B and g : B โ C. Then g โฆ f is a function from A to C and
Dom(g โฆ f) = Dom(f) = A.
Proof.
(i) It is known that Dom(g โฆ f) โ A. Let x โ A. Since f is a function
there exist b โ B such that (a, b) โ f.Also, since g is a function there
is c โ C such that (b, c) โ g. But then (a, c) โ g โฆ f and
a โ Dom(g โฆ f). Thus Dom(g โฆ f) = A.
(ii) Assume that (a, y) and (a, z) are in g โฆ f. Then by definition
SQU-Math2350 4.2 Constructions of Functions Fall 2019 9 / 12
10. Composition of functions: Examples
Example 2.3
Let f and g be real valued functions given by f(x) = sin x and g(x) = ex
. Then
domain of f and g are R. Thus
1 g โฆ f = . . . and Dom(g โฆ f) = . . .
2 f โฆ g = . . . and Dom(f โฆ g) = . . .
Example 2.4
Let f, g and h be real valued functions defined by f(x) = x + 2 and g(x) = 1
xโ9
and h(x) =
โ
x + 5. Then the domain of f, g and h are different. Thus
1 g โฆ f = . . . and Dom(g โฆ f) = . . .. Since Dom(g) = R โ {9}.
2 h โฆ f = . . . and Dom(h โฆ f) = . . .. Since Dom(h) = .....
SQU-Math2350 4.2 Constructions of Functions Fall 2019 10 / 12
11. Theorems on Composition of Functions
Theorem 2.5
Let f : A โ B, g : B โ C and h : C โ D. Then h โฆ (g โฆ f) = (h โฆ g) โฆ f.
Theorem 2.6
Let f : A โ B with Rng(f) = C. If fโ1 is a function then fโ1 โฆ f = IA
and f โฆ fโ1 = IC.
Proof.
Suppose f : A โ B and fโ1 is a function. Then
(i) Dom(fโ1 โฆ f) = Dom(f) = A = Dom(IA).
(ii) For x โ A, there is y โ B such that (x, y) โ f. Then (y, x) โ fโ1.
Thus fโ1 โฆ f(x) = x = IA(x).
From (i) and (ii), fโ1 โฆf = IA. The proof of f โฆfโ1 = IC left as Exercise
SQU-Math2350 4.2 Constructions of Functions Fall 2019 11 / 12
12. Theorems on Composition of Functions
Theorem 2.7
Let f : A โ B. Then f โฆ IA = f and IB โฆ f = f.
Proof.
SQU-Math2350 4.2 Constructions of Functions Fall 2019 12 / 12