The document discusses functions and relations. It defines a function as a special type of relation where each element in the domain is uniquely mapped to an element in the codomain. The document outlines different types of functions, including one-to-one, onto, and bijective functions. It also discusses composing functions and finding the inverse of a function. The overall purpose is to classify and describe different properties and operations related to functions and relations in mathematics.
The document discusses the concept of derivatives. It defines the derivative as the instantaneous rate of change of a function and explains that differentiation is the process of determining the derivative from a given function. It also introduces some key terms used in calculus like d/dx and integral signs. The document goes over basic properties and formulas for derivatives and provides examples of applying the chain rule to find derivatives of some specific functions.
This document discusses key concepts in set theory taught in an Applied Math course, including:
1) Subsets, supersets, proper subsets and proper supersets using subset symbols ⊆, ⊇, ⊂, and ⊃.
2) Cardinality and the power set of a set, and how the cardinality of a power set is 2 to the power of the cardinality of the original set.
3) Venn diagrams and how they can visually represent relationships between sets.
4) The inclusion-exclusion principle for finding the number of elements in the union of two sets.
5) Cartesian products and how they allow sets to be combined.
The document discusses set theory and its applications. It defines what a set is, how sets can be represented, and common set operations like union, intersection, difference and complement. It provides examples to demonstrate set notation and how Venn diagrams can be used to visualize set relationships and operations like DeMorgan's laws. The objectives are to understand set definitions and properties, representation methods, operations and how sets are used in computer science applications.
This document provides an overview of key concepts in entity-relationship modeling including entities, attributes, relationships, constraints, weak entities, and class hierarchies. It defines entities as objects in the real world that can be uniquely identified by a set of attributes. Relationships associate entities and can be one-to-one, one-to-many, many-to-one, or many-to-many. Constraints specify how entities can participate in relationships. Weak entities cannot be uniquely identified without attributes from a related strong entity. Class hierarchies allow entities to be classified into subclasses that inherit attributes.
This document provides an overview of relational database design and normalization. It discusses the goals of database design as generating schemas without unnecessary redundancy and allowing easy data retrieval. Normalization aims to design schemas in a desirable normal form, such as Boyce-Codd normal form (BCNF) or third normal form (3NF). The document introduces key concepts like functional dependencies, normal forms, decomposition, and closure of functional dependencies, which are used to determine if a schema is properly normalized and how to decompose schemas if necessary.
In software engineering, an entity–relationship model (ER model) is a data model for describing the data or information aspects of a business domain or its process requirements
The document discusses inverse functions and one-to-one functions. It defines an inverse function as flipping the inputs and outputs of a function. A function is one-to-one if its inverse is also a function. The horizontal line test can be used to determine if a function is one-to-one, where a one-to-one function only intersects each horizontal line at most once. Composing a function with its inverse results in the identity function.
The document discusses the concept of derivatives. It defines the derivative as the instantaneous rate of change of a function and explains that differentiation is the process of determining the derivative from a given function. It also introduces some key terms used in calculus like d/dx and integral signs. The document goes over basic properties and formulas for derivatives and provides examples of applying the chain rule to find derivatives of some specific functions.
This document discusses key concepts in set theory taught in an Applied Math course, including:
1) Subsets, supersets, proper subsets and proper supersets using subset symbols ⊆, ⊇, ⊂, and ⊃.
2) Cardinality and the power set of a set, and how the cardinality of a power set is 2 to the power of the cardinality of the original set.
3) Venn diagrams and how they can visually represent relationships between sets.
4) The inclusion-exclusion principle for finding the number of elements in the union of two sets.
5) Cartesian products and how they allow sets to be combined.
The document discusses set theory and its applications. It defines what a set is, how sets can be represented, and common set operations like union, intersection, difference and complement. It provides examples to demonstrate set notation and how Venn diagrams can be used to visualize set relationships and operations like DeMorgan's laws. The objectives are to understand set definitions and properties, representation methods, operations and how sets are used in computer science applications.
This document provides an overview of key concepts in entity-relationship modeling including entities, attributes, relationships, constraints, weak entities, and class hierarchies. It defines entities as objects in the real world that can be uniquely identified by a set of attributes. Relationships associate entities and can be one-to-one, one-to-many, many-to-one, or many-to-many. Constraints specify how entities can participate in relationships. Weak entities cannot be uniquely identified without attributes from a related strong entity. Class hierarchies allow entities to be classified into subclasses that inherit attributes.
This document provides an overview of relational database design and normalization. It discusses the goals of database design as generating schemas without unnecessary redundancy and allowing easy data retrieval. Normalization aims to design schemas in a desirable normal form, such as Boyce-Codd normal form (BCNF) or third normal form (3NF). The document introduces key concepts like functional dependencies, normal forms, decomposition, and closure of functional dependencies, which are used to determine if a schema is properly normalized and how to decompose schemas if necessary.
In software engineering, an entity–relationship model (ER model) is a data model for describing the data or information aspects of a business domain or its process requirements
The document discusses inverse functions and one-to-one functions. It defines an inverse function as flipping the inputs and outputs of a function. A function is one-to-one if its inverse is also a function. The horizontal line test can be used to determine if a function is one-to-one, where a one-to-one function only intersects each horizontal line at most once. Composing a function with its inverse results in the identity function.
Dokumen tersebut membahas tentang pengantar teknik informatika yang mencakup penjelasan mengenai flowchart, diagram alir data (DFD), dan komponen-komponennya seperti terminator, proses, dan data store.
This document discusses anti-derivatives and integration. It begins by defining anti-derivatives as the process of determining the function from a given derivative, and defines integration as determining the anti-derivative. It then covers basic properties of integration, methods for evaluating integrals including substitution and integration by parts, and defines definite integration over an interval. Examples are provided to illustrate each concept.
Prosedur dan fungsi digunakan untuk mempermudah pembuatan program dengan cara
mendefinisikan langkah-langkah tertentu sebagai prosedur atau mendefinisikan transformasi nilai
sebagai fungsi. Kedua konsep ini memiliki parameter formal dan aktual yang harus sesuai antara
yang dipanggil dan yang dipanggil. Fungsi menghasilkan nilai sedangkan prosedur dapat
mengubah nilai berdasarkan parameter input/output.
Modul ini membahas tentang deklarasi program dalam bahasa Pascal, meliputi tujuan, dasar teori seperti unsur-unsur program Pascal, tipe data, variabel, dan latihan soal.
Modul ini membahas tentang input dan output data dalam bahasa Pascal. Terdapat pernyataan READ dan WRITE untuk memasukkan dan menampilkan data, serta statement-statement seperti READLN, WRITELN, dan DELAY yang memiliki fungsi tertentu. Modul ini juga memberikan latihan-latihan untuk menerapkan konsep input output data dalam pembuatan program.
Matematika terapan week 5 [compatibility mode]nellylawar
This document discusses functions and relations. It begins by defining a function as a special type of relation where each element of the domain has a single corresponding element in the codomain. The document then outlines objectives about functions, provides definitions of one-to-one, onto, and bijective functions. It also discusses composition of functions, and function inversion. Examples are provided throughout to illustrate these concepts of functions and relations.
1. The document discusses the concepts of relations and functions in mathematics. It defines a function as a unique mapping between elements of two sets.
2. The document covers different types of functions including one-to-one, onto, and bijective functions. It also discusses function composition and inversion.
3. Examples are provided to illustrate function properties, composition, and inversion. The document is intended to help students understand key concepts about relations and functions.
Dokumen tersebut membahas tentang pengantar teknik informatika yang mencakup penjelasan mengenai flowchart, diagram alir data (DFD), dan komponen-komponennya seperti terminator, proses, dan data store.
This document discusses anti-derivatives and integration. It begins by defining anti-derivatives as the process of determining the function from a given derivative, and defines integration as determining the anti-derivative. It then covers basic properties of integration, methods for evaluating integrals including substitution and integration by parts, and defines definite integration over an interval. Examples are provided to illustrate each concept.
Prosedur dan fungsi digunakan untuk mempermudah pembuatan program dengan cara
mendefinisikan langkah-langkah tertentu sebagai prosedur atau mendefinisikan transformasi nilai
sebagai fungsi. Kedua konsep ini memiliki parameter formal dan aktual yang harus sesuai antara
yang dipanggil dan yang dipanggil. Fungsi menghasilkan nilai sedangkan prosedur dapat
mengubah nilai berdasarkan parameter input/output.
Modul ini membahas tentang deklarasi program dalam bahasa Pascal, meliputi tujuan, dasar teori seperti unsur-unsur program Pascal, tipe data, variabel, dan latihan soal.
Modul ini membahas tentang input dan output data dalam bahasa Pascal. Terdapat pernyataan READ dan WRITE untuk memasukkan dan menampilkan data, serta statement-statement seperti READLN, WRITELN, dan DELAY yang memiliki fungsi tertentu. Modul ini juga memberikan latihan-latihan untuk menerapkan konsep input output data dalam pembuatan program.
Matematika terapan week 5 [compatibility mode]nellylawar
This document discusses functions and relations. It begins by defining a function as a special type of relation where each element of the domain has a single corresponding element in the codomain. The document then outlines objectives about functions, provides definitions of one-to-one, onto, and bijective functions. It also discusses composition of functions, and function inversion. Examples are provided throughout to illustrate these concepts of functions and relations.
1. The document discusses the concepts of relations and functions in mathematics. It defines a function as a unique mapping between elements of two sets.
2. The document covers different types of functions including one-to-one, onto, and bijective functions. It also discusses function composition and inversion.
3. Examples are provided to illustrate function properties, composition, and inversion. The document is intended to help students understand key concepts about relations and functions.
Matematika terapan week 4. fungsi dan relasi Hardini_HD
This document introduces the topic of relations and functions in mathematics. It defines what a relation is as a subset of the Cartesian product of two sets. It provides examples of different ways to represent relations visually, including arrow diagrams, tables, matrices, and directed graphs. It also defines inverse relations and describes how to compose two relations. The objectives are to understand these core concepts related to relations between sets as a foundation for further topics in the class.
An Entity–relationship model (ER model) describes the structure of a database with the help of a diagram, which is known as Entity Relationship Diagram (ER Diagram). An ER model is a design or blueprint of a database that can later be implemented as a database. The main components of E-R model are: entity set and relationship set
This document discusses functions and how to determine if a relation represents a function. It defines relations and functions, and explains how to identify the domain and range of a relation. It also describes how to use the vertical line test to determine if a graph or equation defines a function. The document demonstrates using function notation to evaluate functions at given x-values and applies the function concept to an example about annual profits of a jeans company.
LESSON1_REPRESENTATIONS OF FUNCTIONS AND RELATIONS.pptxHannaepascua1
This document contains notes and examples from a mathematics lesson on functions and relations. It defines key terms like relation, function, domain, and range. It provides examples of representing relations and functions using ordered pairs, tables, mapping diagrams, and graphs. It discusses how the vertical line test determines if a graph represents a function. Word problems and learning activities are included to help students understand the concepts.
This document defines limits in mathematics and discusses different types of limits. It begins by explaining that a limit describes what a function value approaches as the input gets arbitrarily close to a given number. The document then provides a formal definition and examples of limits, including one-sided limits. It discusses how to evaluate limits of algebraic functions, infinite limits, and limits of polynomial division. Examples are provided throughout to illustrate limit calculations.
1. Dokumen tersebut menjelaskan tentang pemrograman berbasis objek, yang menerapkan konsep abstraksi, pewarisan, enkapsulasi, dan polimorfisme. Konsep-konsep tersebut memungkinkan pembuatan program dengan memecah masalah menjadi objek-objek yang saling berinteraksi.
2. Diberikan contoh klasifikasi objek seperti hewan dan kendaraan beserta atribut dan perilakunya. Kemudian dijelaskan implementasi konsep-konsep
Dokumen tersebut membahas konsep-konsep dasar pemrograman berorientasi objek pada bahasa Delphi seperti class, object, konstruktor, destruktor, pewarisan, pengkapsulan, polimorfisme dan eksepsi serta operator kelas.
The document provides information about the basic components of a computer system. It discusses the monitor, keyboard, mouse, floppy disk drive, CD/DVD drive, computer peripherals, and various computer terminology. Specifically, it defines each component, describes its function and how it interacts with the other parts of the computer. It also explains common computer terms like CPU, hard drive, modem, motherboard, and operating systems.
Bab V membahas perangkat lunak (software) yang merupakan kumpulan instruksi yang menjalankan perangkat keras komputer. Ada dua jenis software utama yaitu software sistem dan software aplikasi. Software sistem seperti sistem operasi dan driver perangkat bertugas menghubungkan hardware dan pengguna, sedangkan software aplikasi digunakan langsung oleh pengguna. Linux dan Windows adalah contoh sistem operasi paling populer yang masing-masing memiliki kelebihan tersendiri.
Dokumen membahas tentang sistem bilangan yang digunakan dalam komputer seperti sistem bilangan biner, oktal, desimal, dan heksadesimal beserta penjelasan konversi antar sistem bilangan tersebut."
Dokumen tersebut membahas tentang konsep jaringan komputer, komponen-komponen jaringan seperti node dan link, topologi jaringan seperti bus, bintang, cincin dan mesh, serta jenis-jenis jaringan berdasarkan distribusi sumber daya, jangkauan geografis, hubungan antar komputer, dan media transmisi.
The document provides an introduction to operating systems, including their definition, roles, functions, and history. It defines an operating system as the software that manages hardware and allows it to be usable. Operating systems manage system resources, files, memory, processors, and input/output. Examples of common operating systems include Windows, Mac OS, Unix, and Linux. The document then outlines the evolution of operating systems from early vacuum tube computers through personal computers, and describes basic operating system concepts like processes, address spaces, files, and protection.
This document provides an overview of software, including its history, types, and evolution. It discusses how software allows computers to compute and process data through program instructions. It defines software as collections of computer programs and related data that provide instructions for what a computer should do. It also describes the different levels of software processing and the major eras in software evolution from pioneers to modern times. Finally, it outlines the main types of software, differentiating between system software that manages computer systems and application software for specific tasks.
Mobile internet represents a new computing cycle that is led by non-US markets. It provides uncharted sources of usage and potential killer applications involving social networking, video, location-based services, and more. Factors driving mobile internet adoption include handsets becoming small computers, cheaper and faster data/networks, and more available content. Within two years, wireless consumers expect always-on superfast access to nearly all information on elegant portable devices with day-long battery life.
This document provides an overview of the history and evolution of the Internet. It discusses how the Internet originated from early computer networks developed by ARPA and DARPA in the 1960s-1970s to connect government and university research computers. It then covers the creation of TCP/IP in the 1970s which established the fundamental communication protocols of the Internet. The document also summarizes the commercial opening of the Internet in the 1990s and the creation of the World Wide Web in 1989-1991 by Tim Berners-Lee, which allowed for easy access to hyperlinked documents and multimedia over the Internet.
This document provides an overview of key concepts related to e-government. It discusses how governments have increasingly adopted information and communication technologies (ICTs) and how e-government can be defined. E-government is described as a tool to enable better policy outcomes, higher quality services, and greater efficiency and engagement with citizens. The document also outlines some of the main reasons for governments to pursue e-government, including improved efficiency and service delivery, as well as challenges that can be faced, such as legislative and budgetary barriers, technology changes, and addressing the digital divide.
Teks tersebut membahas perangkat keras sistem utama komputer, termasuk motherboard, CPU, memori, dan media penyimpanan. Juga membahas sejarah dan perkembangan dari masing-masing komponen tersebut.
1. This chapter defines electronic commerce and describes its various categories. It outlines the content and framework of EC and discusses the major types of EC transactions.
2. The digital revolution and changing business environment are described as key drivers of the growth of EC. Pressure from markets, society, and technology are pushing organizations to adopt EC strategies.
3. The benefits of EC to organizations, consumers, and society are discussed, along with some limitations of EC. Common EC business models and how EC can help organizations respond to environmental pressures are also summarized.
Dokumen tersebut membahas tentang sistem bilangan dan format data yang digunakan dalam sistem digital, meliputi sistem bilangan desimal, biner, oktal dan heksadesimal beserta konversi antara sistem-sistem bilangan tersebut. Dokumen tersebut juga menjelaskan organisasi data dalam bit, nibble, byte dan word pada komputer.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"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 Analytics” 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!"
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
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Film vocab for eal 3 students: Australia the movie
Matematika terapan week 5
1. 1
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 4
Relation and Function II
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
Overview
In mathematics, function is a relation. A function
establishes or expresses the “relation”-ship
between objects. In computer systems, for
instance, the input is fed to the system in form of
data or objects and the system generates the
output that will be the function of input. So, in other
words, function is the mapping or transformation of
objects from one form to other.
In this section we will concentrate our discussion
on function and its classifications.
2. 2
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
Objectives
Definition of Function
Function Properties
Composition of Function
Function Inversion
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
Definition of Function
As mention above, function is a relation.
However, this definition cannot be
interchanged because the function has an
unique relationship.
Let A and B is non-empty sets. A relation
from A to B is said as a function if all
element of A has only “one connection” to B.
3. 3
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
Some Functions
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
Not a Function
4. 4
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
A function A to B can be written as :
f : A B
A and B is called as Domain and Codomain
respectively.
We also can write it as
f(a) = b,
where a∈A and b∈B
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
Denote all elements into a function form!!!
.3
.w
.a
.7
.#
.s
.4
.&
.r
.10
5. 5
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
Frequently, function is expressed in
mathematical formulas. Example:
Find the codomain of f(x) = 2x+3 for 1≤x≤ 4,
x∈Z. Draw the graph as well.
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function
Find the domains and codomains from the
graph f(x) = x2+2 below !!
6. 6
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Function Properties
There are three basic type of function, those
are :
1. One-to-one function (injective)
2. Onto function (surjective)
3. Bijective function (both one-to-one and onto)
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Injective Function
A function f: A B is said to be one-to-one
(written 1-1) if different elements in the domain A
have distinct match in the codomain B.
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Surjective Function
A function f: A B is said to be an onto function if
each element of codomain B is the image of some
element of domain A.
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Bijective Function
Bijective can be called correspondence one to
one. It can be inverted.
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Composition of Function
Let g be a function from the set A to the set B and
let f be a function from the set B to the set C. The
composition of the functions f and g, denoted by
f o g, is defined by
(f o g)(a) = f(g(a)).
Therefore, to find (f o g)(a) we first apply the
function g to a to obtain g(a) and then we apply the
function f to the result g(a) to obtain (f o g)(a) =
f(g(a)).
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Ex.
Let g be the function from the set (a, b, c} to itself
such that g(a) = b, g(b) = c, and g(c) = a. Let f be
the function from the set {a, b, c} to the set {1, 2, 3}
such that f(a) = 3, f(b) = 2, and f(c) = 1. What is the
composition of f and g?
Solution :
The composition f o g is defined by (f o g)(a) =
f(g(a)) = f(b) = 2, (f o g) (b) = f(g(b)) = f(c) = 1, and
(f o g)(c) = f(g(c)) = f(a) = 3.
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Exercise
Let f and g be the functions from the set of integers
to the set of integers defined by f(x) = 2x + 3 and
g(x) = 3x + 2. For x = 2 and -2, what is the
composition of f and g? What is the composition of
g and f?
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Function Inversion
Let f be a one-to-one correspondence from the set
A to the set B. The inverse function of f is the
function that assigns to an element b belonging to
B the unique element a in A such that f(a) = b.
The inverse function of f is denoted by f-1. Hence,
f-1(b)= a when f(a) = b.
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Ex.
Let f be the function from {a, b, c} to {1, 2, 3} such
that f(a) = 2, f(b) = 3, and f(c) = 1. Is the f
invertible? If it is, what is its inverse?
Solution:
The function f is invertible because it is a one-to-
one correspondence. The invers function f
reverses the correspondence given by f, so
f-1(1) = c, f -1(2) = a, and f-1(3)=b.
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For mathematics formulas, to find invers of f,
we can exchange the variable x with y.
Ex.