This document provides a summary of a lecture on teaching discrete mathematics. It discusses key topics in discrete mathematics including discrete objects, binary and unary operations, and algebraic structures. It also covers Boolean matrices, modular arithmetic, mathematical logic, and truth tables. Examples are given for different concepts like binary operations on sets of integers and Fibonacci numbers. The document aims to train teachers on effectively teaching discrete mathematics concepts.
This document contains the content of a lecture delivered to train teachers on how to teach applications of derivatives. The lecture covers various topics related to teaching including learning habits, teaching habits, strategies for being an effective teacher and mathematics teacher. It also discusses specific applications of derivatives like rates of change, related rates problems, maxima and minima, and indeterminate forms. Examples are provided throughout to illustrate key concepts. The overall summary is that the lecture aimed to equip teachers with effective strategies and content knowledge to teach applications of derivatives.
This document contains a daily lesson log for a 7th grade mathematics class. The lesson covers algebraic expressions, properties of real numbers, linear equations, and inequalities in one variable. The lesson objectives are to differentiate between equations and inequalities, illustrate linear equations and inequalities, and find solutions to linear equations and inequalities. The lesson content includes differentiating equations and inequalities, linear equations and inequalities in one variable, and solving linear equations and inequalities. Learning resources and procedures are outlined for reviewing concepts, examples, practice, and application. Formative assessments are used to check student understanding.
Lesson 5 introduces exponents to students. Students learn that exponents represent repeated multiplication of the same factor. Key terms introduced are base, exponent or power, squared, and cubed. Students practice writing expressions in exponential, expanded, and standard form. They learn that when the exponent is 2, it is called squared, and when the exponent is 3, it is called cubed. The lesson reinforces the relationship between exponents and repeated multiplication.
The document provides an overview of Unit 1 of the third grade mathematics curriculum for the Isaac School District. It focuses on representing numbers in different ways and relationships between numbers. The unit covers addition, subtraction, and place value over 16 sessions. It includes essential questions, vocabulary, Arizona state standards, and examples for how students can explore numerical patterns and operations with multiples of 10 to develop fluency.
The document is a lecture on counting principles including permutations, combinations, and the pigeonhole principle. It begins by explaining the product rule and sum rule for counting. It then defines permutations as ordered arrangements and combinations as unordered selections. Permutations of a set are denoted by P(n,r) and combinations by C(n,r). Examples are provided to illustrate permutations, combinations, and the pigeonhole principle.
This document summarizes key concepts from a lecture on decidability in formal language theory:
- Context-free languages (CFLs) are decidable as their acceptability can be determined by a Turing machine that simulates a pushdown automaton.
- The acceptability problem for context-free grammars (ACFG) - determining if a string is in the language of a given CFG - is decidable using a algorithm that lists all derivations.
- This shows that the class of languages recognized by Turing machines is decidable.
- However, diagonalization arguments show that some problems are undecidable, such as the halting problem of determining if an arbitrary Turing machine
Scientific notation is used to write very large or small numbers in a standardized way. It takes the form of a number between 1 and 10 multiplied by a power of 10. This makes it easier to work with numbers that scientists and engineers commonly use. Standard form is the normal way we write numbers, while scientific notation moves the decimal place to have a leading digit between 1 and 9, and adds an exponent of 10 to show how many places the decimal was moved. This allows even huge and tiny numbers to be expressed concisely for calculations and comparisons.
I am Simon M. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Texas, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
This document contains the content of a lecture delivered to train teachers on how to teach applications of derivatives. The lecture covers various topics related to teaching including learning habits, teaching habits, strategies for being an effective teacher and mathematics teacher. It also discusses specific applications of derivatives like rates of change, related rates problems, maxima and minima, and indeterminate forms. Examples are provided throughout to illustrate key concepts. The overall summary is that the lecture aimed to equip teachers with effective strategies and content knowledge to teach applications of derivatives.
This document contains a daily lesson log for a 7th grade mathematics class. The lesson covers algebraic expressions, properties of real numbers, linear equations, and inequalities in one variable. The lesson objectives are to differentiate between equations and inequalities, illustrate linear equations and inequalities, and find solutions to linear equations and inequalities. The lesson content includes differentiating equations and inequalities, linear equations and inequalities in one variable, and solving linear equations and inequalities. Learning resources and procedures are outlined for reviewing concepts, examples, practice, and application. Formative assessments are used to check student understanding.
Lesson 5 introduces exponents to students. Students learn that exponents represent repeated multiplication of the same factor. Key terms introduced are base, exponent or power, squared, and cubed. Students practice writing expressions in exponential, expanded, and standard form. They learn that when the exponent is 2, it is called squared, and when the exponent is 3, it is called cubed. The lesson reinforces the relationship between exponents and repeated multiplication.
The document provides an overview of Unit 1 of the third grade mathematics curriculum for the Isaac School District. It focuses on representing numbers in different ways and relationships between numbers. The unit covers addition, subtraction, and place value over 16 sessions. It includes essential questions, vocabulary, Arizona state standards, and examples for how students can explore numerical patterns and operations with multiples of 10 to develop fluency.
The document is a lecture on counting principles including permutations, combinations, and the pigeonhole principle. It begins by explaining the product rule and sum rule for counting. It then defines permutations as ordered arrangements and combinations as unordered selections. Permutations of a set are denoted by P(n,r) and combinations by C(n,r). Examples are provided to illustrate permutations, combinations, and the pigeonhole principle.
This document summarizes key concepts from a lecture on decidability in formal language theory:
- Context-free languages (CFLs) are decidable as their acceptability can be determined by a Turing machine that simulates a pushdown automaton.
- The acceptability problem for context-free grammars (ACFG) - determining if a string is in the language of a given CFG - is decidable using a algorithm that lists all derivations.
- This shows that the class of languages recognized by Turing machines is decidable.
- However, diagonalization arguments show that some problems are undecidable, such as the halting problem of determining if an arbitrary Turing machine
Scientific notation is used to write very large or small numbers in a standardized way. It takes the form of a number between 1 and 10 multiplied by a power of 10. This makes it easier to work with numbers that scientists and engineers commonly use. Standard form is the normal way we write numbers, while scientific notation moves the decimal place to have a leading digit between 1 and 9, and adds an exponent of 10 to show how many places the decimal was moved. This allows even huge and tiny numbers to be expressed concisely for calculations and comparisons.
I am Simon M. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Texas, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
This document provides materials for a training session on English related to mathematics for mathematics teachers. It covers three units:
Unit 1 introduces the purpose of learning English for mathematics teachers and the objectives of the training session.
Unit 2 explains concepts related to numbers in English, including types of numbers, mathematical operations, fractions, decimals, percentages and ratios. Examples and exercises are provided.
Unit 3 discusses ratios, scales, proportions, percentages and interest in financial contexts. It provides more examples and exercises for participants to practice.
The materials are designed for independent study but also for use during the training session, where participants can ask questions. The overall goal is to help mathematics teachers improve their English comprehension of mathematics literature.
This document provides an overview of the topics that will be covered in a finite mathematics course, including residue arithmetic, elements of finite groups/rings/fields, number theory concepts like the Euclidean algorithm and Chinese Remainder Theorem, and basics of finite vector spaces and fields. The style of the course will be leisurely and discursive, focusing on mathematical thinking and discovery. While the mathematics is classical, it will be new to students. The goal is to emphasize elegance and aesthetics over utility alone.
This document provides an introduction to discrete mathematics. It defines discrete objects as those that are distinct and separated from each other, such as integers. It explains that discrete mathematics deals with discrete objects and their properties. The document also outlines some key concepts that will be covered, including sets, functions, graphs, logic, and proofs. It discusses how formal approaches in discrete math can help handle large or indefinite quantities and provide reusable results.
This lesson teaches students about the relationship between division and subtraction. Students use tape diagrams to model division problems and represent them as repeated subtraction. They discover that if 12 is divided by x equals 4, then 12 can be written as a subtraction sentence where x is subtracted 4 times until reaching 0. The key relationship students learn is that the number of times the divisor is subtracted from the dividend is the same as the quotient. Students practice modeling more examples to strengthen their understanding of this important mathematical relationship.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Aenean commodo ligula eget dolor. Aenean massa. Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Donec quam felis, ultricies nec, pellentesque eu, pretium quis, sem. Nulla consequat massa quis enim. Donec pede justo, fringilla vel, aliquet nec, vulputate eget, arcu. In enim justo, rhoncus ut, imperdiet a, venenatis vitae, justo. Nullam dictum felis eu pede mollis pretium. Integer tincidunt. Cras dapibus. Vivamus elementum semper nisi. Aenean vulputate eleifend tellus. Aenean leo ligula, porttitor eu, consequat vitae, eleifend ac, enim. Aliquam lorem ante, dapibus in, viverra quis, feugiat a, tellus. Phasellus viverra nulla ut metus varius laoreet. Quisque rutrum. Aenean imperdiet. Etiam ultricies nisi vel augue. Curabitur ullamcorper ultricies nisi. Nam eget dui. Etiam rhoncus. Maecenas tempus, tellus eget condimentum rhoncus, sem quam semper libero, sit amet adipiscing sem neque sed ipsum. Nam quam nunc, blandit vel, luctus pulvinar, hendrerit id, lorem. Maecenas nec odio et ante tincidunt tempus. Donec vitae sapien ut libero venenatis faucibus. Nullam quis ante. Etiam sit amet orci eget eros faucibus tincidunt. Duis leo. Sed fringilla mauris sit amet nibh. Donec sodales sagittis magna. Sed consequat, leo eget bibendum sodales, augue velit cursus nunc,
This document outlines several recursive algorithms and discusses their recursive definitions, correctness proofs, and running times. It provides examples of using recursion to define functions that compute sums and products. The binary search algorithm is presented recursively and its correctness is proved by strong induction on the size of the input list. Recursive definitions and notation are introduced for sums and products. The running time of binary search is shown to be logarithmic in the size of the input.
The document provides information about linear and nonlinear expressions including:
- Definitions of linear and nonlinear expressions, where linear expressions can only have x terms with an exponent of 1 and nonlinear expressions have exponents not equal to 1 or 0.
- Examples of sorting expressions into linear and nonlinear groups and explaining the reasoning. Linear expressions follow the definition while nonlinear have exponents not 1 or 0.
- The importance of distinguishing between linear and nonlinear expressions is because students will soon learn to solve linear equations, with nonlinear equations addressed later. It also relates to predicting graph shapes.
- Additional examples analyze expressions and identify them as linear or nonlinear based on the definitions and exponents of x terms. Students are instructed to practice identifying expressions
Students will be taking standardized tests from October 26th to 30th so there will be no live classes during that time. The document provides definitions and examples to help students distinguish between linear and nonlinear expressions. Linear expressions can only have variables raised to the power of 1 or 0, while nonlinear expressions have variables raised to other powers. Being able to identify linear and nonlinear expressions is important because students will be learning to solve linear equations, and want to predict the shapes of their graphs. A series of examples are provided to help students apply the definitions.
The document provides an outline for a teaching week on functional mathematics for year 7 students. It includes topics such as numbers, fractions, decimals, measurement, and area/perimeter. The topics and outcomes are listed along with instructional approaches and strategies using examples, activities, and resources to explain key concepts in numbers, operations, and measurement.
The document provides an overview of topics and learning outcomes for a 7th grade functional mathematics teaching week. It includes instruction on:
1. Numbers - reading, writing, representing, and comparing numbers up to millions.
2. Fractions - representing, comparing, adding, and subtracting fractions.
3. Combined operations - performing multi-step calculations involving addition, subtraction, multiplication, and division.
The document outlines instructional approaches and strategies as well as recommended resources for teaching each topic.
Writing and solving equations can be abstract and confusing for students. Learn nonconventional ways to encourage flexible thinking and develop a deeper understanding of inverse relationships, fact families, and variables representation. Walk away with three easy-to-use activities to expand students' toolkit for solving equations.
This document is a daily lesson log for a Grade 1 mathematics class. It outlines the objectives, content, learning resources and procedures for five days of lessons on division of whole numbers up to 1000 and unit fractions. The lessons include visualizing and representing division using equal sharing, repeated subtraction, number lines and equal groups. Formative assessments are conducted to evaluate student learning and additional activities are provided for remediation.
The document discusses finite automata and theory of computation. It begins by defining finite automata as abstract machines that can have a finite number of states and read finite input strings, as well as the basic components of an automaton. It then explains the differences between deterministic finite automata (DFAs) and non-deterministic finite automata (NFAs), and provides examples of transition diagrams for visualizing state transitions in automata. The document aims to introduce the basics of finite automata as part of the theory of computation.
This document discusses mathematics teaching planning and examples of content knowledge, pedagogical content knowledge, diversity responsive curriculum, and levels of understanding in teaching mathematics. It provides examples for declarative, procedural, and pedagogical knowledge. It also gives examples for creating an inclusive curriculum, connecting content to students' lives, and analyzing subject matter, concepts, principles, and tasks in mathematics teaching.
The document contains a student's reflections on their IGCSE mathematics course taught in English. The student discusses gaining knowledge of math terms in English and hopes to one day teach math in English. They note some difficulty understanding when presentations are fully in English but can understand with Indonesian translations. The student hopes the IGCSE maths course continues to enhance learning and that students and faculty actively use English in class to improve language fluency.
This lesson plan is for a 9th standard mathematics class on irrational numbers. The teacher will help students understand irrational numbers through discussion and activities. Students will learn that integers and fractions are rational numbers, while some numbers like the square root of 2 cannot be expressed as fractions and are called irrational numbers. Through examples and working through proofs, students will understand the key concepts and be able to identify and provide examples of irrational numbers.
This document provides a daily lesson log for a Grade 8 mathematics class taught by Luisa F. Garcillan. The lesson covers linear inequalities in two variables over four class sessions. The objectives are to illustrate, differentiate, and graph linear inequalities in two variables. References and learning resources include textbooks, guides, and online materials. The procedures involve reviewing concepts, presenting examples, discussing new concepts, and practicing skills through activities, graphing, and word problems. The goal is for students to understand and be able to solve problems involving linear inequalities in two variables.
This lesson teaches students about the relationship between addition and subtraction through the use of tape diagrams and algebraic expressions. Students explore identities such as w - x + x = w by building tape diagrams that represent adding and then subtracting the same amount. They recognize that regardless of the numbers used, adding a number and then subtracting the same number will result in the original amount. Students practice writing number sentences to represent identities and evaluating expressions that involve addition and subtraction of the same terms. The lesson aims to help students understand that identities will always be true no matter what numbers are substituted in.
The document provides an overview of Unit 5 of the 4th grade mathematics curriculum for the Isaac School District. The unit focuses on addition, subtraction, and place value with large numbers up to 1,000,000. It includes 27 instructional sessions covering key concepts like using place value to represent numbers in expanded form and standard algorithms for addition and subtraction. Students are expected to fluently perform multi-digit calculations and explain their work by referring to place value. The unit vocabulary and Arizona state math standards addressed are also outlined.
This document provides materials for a training session on English related to mathematics for mathematics teachers. It covers three units:
Unit 1 introduces the purpose of learning English for mathematics teachers and the objectives of the training session.
Unit 2 explains concepts related to numbers in English, including types of numbers, mathematical operations, fractions, decimals, percentages and ratios. Examples and exercises are provided.
Unit 3 discusses ratios, scales, proportions, percentages and interest in financial contexts. It provides more examples and exercises for participants to practice.
The materials are designed for independent study but also for use during the training session, where participants can ask questions. The overall goal is to help mathematics teachers improve their English comprehension of mathematics literature.
This document provides an overview of the topics that will be covered in a finite mathematics course, including residue arithmetic, elements of finite groups/rings/fields, number theory concepts like the Euclidean algorithm and Chinese Remainder Theorem, and basics of finite vector spaces and fields. The style of the course will be leisurely and discursive, focusing on mathematical thinking and discovery. While the mathematics is classical, it will be new to students. The goal is to emphasize elegance and aesthetics over utility alone.
This document provides an introduction to discrete mathematics. It defines discrete objects as those that are distinct and separated from each other, such as integers. It explains that discrete mathematics deals with discrete objects and their properties. The document also outlines some key concepts that will be covered, including sets, functions, graphs, logic, and proofs. It discusses how formal approaches in discrete math can help handle large or indefinite quantities and provide reusable results.
This lesson teaches students about the relationship between division and subtraction. Students use tape diagrams to model division problems and represent them as repeated subtraction. They discover that if 12 is divided by x equals 4, then 12 can be written as a subtraction sentence where x is subtracted 4 times until reaching 0. The key relationship students learn is that the number of times the divisor is subtracted from the dividend is the same as the quotient. Students practice modeling more examples to strengthen their understanding of this important mathematical relationship.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Aenean commodo ligula eget dolor. Aenean massa. Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Donec quam felis, ultricies nec, pellentesque eu, pretium quis, sem. Nulla consequat massa quis enim. Donec pede justo, fringilla vel, aliquet nec, vulputate eget, arcu. In enim justo, rhoncus ut, imperdiet a, venenatis vitae, justo. Nullam dictum felis eu pede mollis pretium. Integer tincidunt. Cras dapibus. Vivamus elementum semper nisi. Aenean vulputate eleifend tellus. Aenean leo ligula, porttitor eu, consequat vitae, eleifend ac, enim. Aliquam lorem ante, dapibus in, viverra quis, feugiat a, tellus. Phasellus viverra nulla ut metus varius laoreet. Quisque rutrum. Aenean imperdiet. Etiam ultricies nisi vel augue. Curabitur ullamcorper ultricies nisi. Nam eget dui. Etiam rhoncus. Maecenas tempus, tellus eget condimentum rhoncus, sem quam semper libero, sit amet adipiscing sem neque sed ipsum. Nam quam nunc, blandit vel, luctus pulvinar, hendrerit id, lorem. Maecenas nec odio et ante tincidunt tempus. Donec vitae sapien ut libero venenatis faucibus. Nullam quis ante. Etiam sit amet orci eget eros faucibus tincidunt. Duis leo. Sed fringilla mauris sit amet nibh. Donec sodales sagittis magna. Sed consequat, leo eget bibendum sodales, augue velit cursus nunc,
This document outlines several recursive algorithms and discusses their recursive definitions, correctness proofs, and running times. It provides examples of using recursion to define functions that compute sums and products. The binary search algorithm is presented recursively and its correctness is proved by strong induction on the size of the input list. Recursive definitions and notation are introduced for sums and products. The running time of binary search is shown to be logarithmic in the size of the input.
The document provides information about linear and nonlinear expressions including:
- Definitions of linear and nonlinear expressions, where linear expressions can only have x terms with an exponent of 1 and nonlinear expressions have exponents not equal to 1 or 0.
- Examples of sorting expressions into linear and nonlinear groups and explaining the reasoning. Linear expressions follow the definition while nonlinear have exponents not 1 or 0.
- The importance of distinguishing between linear and nonlinear expressions is because students will soon learn to solve linear equations, with nonlinear equations addressed later. It also relates to predicting graph shapes.
- Additional examples analyze expressions and identify them as linear or nonlinear based on the definitions and exponents of x terms. Students are instructed to practice identifying expressions
Students will be taking standardized tests from October 26th to 30th so there will be no live classes during that time. The document provides definitions and examples to help students distinguish between linear and nonlinear expressions. Linear expressions can only have variables raised to the power of 1 or 0, while nonlinear expressions have variables raised to other powers. Being able to identify linear and nonlinear expressions is important because students will be learning to solve linear equations, and want to predict the shapes of their graphs. A series of examples are provided to help students apply the definitions.
The document provides an outline for a teaching week on functional mathematics for year 7 students. It includes topics such as numbers, fractions, decimals, measurement, and area/perimeter. The topics and outcomes are listed along with instructional approaches and strategies using examples, activities, and resources to explain key concepts in numbers, operations, and measurement.
The document provides an overview of topics and learning outcomes for a 7th grade functional mathematics teaching week. It includes instruction on:
1. Numbers - reading, writing, representing, and comparing numbers up to millions.
2. Fractions - representing, comparing, adding, and subtracting fractions.
3. Combined operations - performing multi-step calculations involving addition, subtraction, multiplication, and division.
The document outlines instructional approaches and strategies as well as recommended resources for teaching each topic.
Writing and solving equations can be abstract and confusing for students. Learn nonconventional ways to encourage flexible thinking and develop a deeper understanding of inverse relationships, fact families, and variables representation. Walk away with three easy-to-use activities to expand students' toolkit for solving equations.
This document is a daily lesson log for a Grade 1 mathematics class. It outlines the objectives, content, learning resources and procedures for five days of lessons on division of whole numbers up to 1000 and unit fractions. The lessons include visualizing and representing division using equal sharing, repeated subtraction, number lines and equal groups. Formative assessments are conducted to evaluate student learning and additional activities are provided for remediation.
The document discusses finite automata and theory of computation. It begins by defining finite automata as abstract machines that can have a finite number of states and read finite input strings, as well as the basic components of an automaton. It then explains the differences between deterministic finite automata (DFAs) and non-deterministic finite automata (NFAs), and provides examples of transition diagrams for visualizing state transitions in automata. The document aims to introduce the basics of finite automata as part of the theory of computation.
This document discusses mathematics teaching planning and examples of content knowledge, pedagogical content knowledge, diversity responsive curriculum, and levels of understanding in teaching mathematics. It provides examples for declarative, procedural, and pedagogical knowledge. It also gives examples for creating an inclusive curriculum, connecting content to students' lives, and analyzing subject matter, concepts, principles, and tasks in mathematics teaching.
The document contains a student's reflections on their IGCSE mathematics course taught in English. The student discusses gaining knowledge of math terms in English and hopes to one day teach math in English. They note some difficulty understanding when presentations are fully in English but can understand with Indonesian translations. The student hopes the IGCSE maths course continues to enhance learning and that students and faculty actively use English in class to improve language fluency.
This lesson plan is for a 9th standard mathematics class on irrational numbers. The teacher will help students understand irrational numbers through discussion and activities. Students will learn that integers and fractions are rational numbers, while some numbers like the square root of 2 cannot be expressed as fractions and are called irrational numbers. Through examples and working through proofs, students will understand the key concepts and be able to identify and provide examples of irrational numbers.
This document provides a daily lesson log for a Grade 8 mathematics class taught by Luisa F. Garcillan. The lesson covers linear inequalities in two variables over four class sessions. The objectives are to illustrate, differentiate, and graph linear inequalities in two variables. References and learning resources include textbooks, guides, and online materials. The procedures involve reviewing concepts, presenting examples, discussing new concepts, and practicing skills through activities, graphing, and word problems. The goal is for students to understand and be able to solve problems involving linear inequalities in two variables.
This lesson teaches students about the relationship between addition and subtraction through the use of tape diagrams and algebraic expressions. Students explore identities such as w - x + x = w by building tape diagrams that represent adding and then subtracting the same amount. They recognize that regardless of the numbers used, adding a number and then subtracting the same number will result in the original amount. Students practice writing number sentences to represent identities and evaluating expressions that involve addition and subtraction of the same terms. The lesson aims to help students understand that identities will always be true no matter what numbers are substituted in.
The document provides an overview of Unit 5 of the 4th grade mathematics curriculum for the Isaac School District. The unit focuses on addition, subtraction, and place value with large numbers up to 1,000,000. It includes 27 instructional sessions covering key concepts like using place value to represent numbers in expanded form and standard algorithms for addition and subtraction. Students are expected to fluently perform multi-digit calculations and explain their work by referring to place value. The unit vocabulary and Arizona state math standards addressed are also outlined.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
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 ...
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
How to Build a Module in Odoo 17 Using the Scaffold Method
Chapter 12.pptx
1. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Jacob(II) Bernoulli
Born: 17 October 1759 in Basel,
Switzerland
Died: 3rd July 1789 in St
Petersburg, Russia
2. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Discrete Objects
• Discrete objects can often be enumerated by
integers
• Examples of discrete objects include buildings,
roads, and parcels.
• Discrete objects are usually nouns.
3. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Discrete Mathematics
• Discrete mathematics is the study of mathematical
structures that are fundamentally discrete rather than
continuous. ...
• Discrete mathematics therefore excludes topics in
"continuous mathematics" such as calculus and
analysis.
• Some of the branches of Discrete Mathematics are
combinatorics, mathematical logic, boolean algebra,
graph theory, coding theory etc.
4. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Unary operation
𝜓: 𝐴 ⟶ 𝐴
𝑇ℎ𝑎𝑡𝑖𝑠∀𝑎, ∈ 𝐴, 𝜓(𝑎) ∈ 𝐴
That is , any function from a non -empty set into itself
is an unary operation .
𝐿𝑒𝑡𝐴𝑏𝑒ℙ(𝑆) − 𝑡ℎ𝑒𝑝𝑜𝑤𝑒𝑟𝑠𝑒𝑡𝑜𝑓𝑆
Then for a subset C of A if we associate the
complement of C in A is an unary operation .
𝐿𝑒𝑡𝐴𝑏𝑒𝕄𝑛×𝑛
Then for a matrix C if we associate the adjoint of
C in is an unary operation .
5. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Binary operations
∗: 𝐴 × 𝐴 ⟶ 𝐴
𝑇ℎ𝑎𝑡𝑖𝑠∀𝑎, 𝑏 ∈ 𝐴, 𝑎 ∗ 𝑏 ∈ 𝐴
⟨𝐴,∗⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
Examples
1)⟨𝑁, +⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
2)⟨𝑁,×⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
3)⟨𝑁, −⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑛𝑜𝑡𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
4)⟨𝑁,/⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑛𝑜𝑡𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
Let N be the set of all Natural numbers
6. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
7. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Algebraic structure
Examples
⟨𝐴,∗⟩𝑖𝑠𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
Let A = Z , Q , R be respectively the set of all integers,
rational , real, complex numbers numbers, then
Check for which arithmetic operation
On the set of all integers verify the operation is binary
⟨𝑍,∗⟩𝑔𝑖𝑣𝑒𝑛𝑏𝑦𝑚 ∗ 𝑛 = 𝑚 + 𝑛 − 𝑚. 𝑛
𝑌𝐸𝑆
8. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Properties of Binary Operation 𝐿𝑒𝑡⟨𝐴,∗⟩𝑏𝑒𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
Sl.No Property If
1 Commutative
2 Associative
3
Existence of
Identity
4
Existence of
Inverse
Identity and inverse if exists then they are unique w.r.to
the binary operation
9. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
on and multiplication satisfies all the properties mentioned earlier.
The additive identity is ‘0 ‘
Inverse for any ’n’ under addition is ‘ - n’
The multiplicative identity is ‘1’
Inverse for any non -zero number ‘x’ is 1/x
𝑎 ∗ 𝑏 = 𝑎 𝑏𝑖𝑠𝑛𝑜𝑡𝑎𝑏𝑖𝑛𝑎𝑟𝑦𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑖𝑛ℝ
𝐵𝑒𝑐𝑎𝑢𝑠𝑒𝑎 ∗ −1 = 𝑎 −1 ∉ ℝ
10. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Is the addition and multiplication, on the set of all
prime numbers , a binary operation ?
Is the addition, on the set of all Fibonacci numbers ,
a binary operation ?
NO because 2 and 13 are Fibonacci numbers but 2+13
= 15 is not so.
NO because
11. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Boolean Matrices
A Boolean matrix is of the form
12. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
• Remark: The operations of Join & Meet defines a binary
operations on the the set of all Boolean matrices .
• Those two binary operations are commutative and
associative which posses identity but no inverse
13. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Operation table for a finite set with binary operation
Each row is a permutation
Binary operation is non -
commutative
Is the Binary operation
associative
c a
a
Fill in the following table so that
the binary operation ∗ on
A = {a, b, c} is commutative
14. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Modulo operations
𝑂𝑛𝑡ℎ𝑒𝑠𝑒𝑡𝑜𝑓𝑎𝑙𝑙𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠ℤ
𝐷𝑒𝑓𝑖𝑛𝑒𝑓𝑜𝑟𝑚, 𝑛 ∈ ℤ𝑚 +𝑝 𝑛𝑎𝑠,
the remainder when m+n is divided by p
𝑇ℎ𝑒𝑛 +𝑝 𝑖𝑠𝑎𝑏𝑖𝑛𝑎𝑟𝑦𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑜𝑛ℤ
⟨ℤ, +𝑝⟩𝑖𝑠𝑑𝑒𝑛𝑜𝑡𝑒𝑑𝑏𝑦𝑍𝑝
15. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Binary operation table for
16. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
International Standard Book Number: (ISBN) &
Modular arithmetic
• Books are identified by a ten digit number, abbreviated by
ISBN.
• For example, the book Field and Galois Theory, published by
Springer, has the number ISBN 0-387-94753-1.
• The first digit identifies the language in which the book is
written,
• The second block of digits identifies the publisher,
• The third block identifies the book itself, and
• The final digit is the check digit.
17. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
• In this scheme, each digit can be a numeral 0,..., 9 or X, which
represents 10. A ten digit number is a valid ISBN
provided that
is divisible by 11
Hence a valid ISBN number
For the validity 10 .0+9.3+8.8+7.7 +6.9+5.7+4.3+3.2+2.9+? = multiple of 11.
That is 265 +? = (11.24 + 1)+? must be divisible by 11
Therefore ? = 10 = X
18. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Mathematical logic
• What is a Logic ?
• What is a Mathematical Logic?
• Logic (from the Greek "logos", has a meanings including word,
thought, idea, argument, account, reason or principle) is
the study of reasoning, or the study of the principles and criteria
of valid inference and demonstration.
• It attempts to distinguish good reasoning from bad reasoning.
• Mathematical logic is often divided into the fields of set
theory, model theory, recursion theory, and proof theory.
19. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
• To build any language we need to have the first alphabet
alphabets , words , statements and valid statements.
• For example we have,
Alphabets Words Language Other languages
அ, ஆ, இ,
ஈ,…
அன் பு தமிழ்
A, B , C ,
D,…
Love English French, German
Αγάπη
Greek
0, 1, 2 , 3,…9 12 15 22 5 Decimal Number Ternary
0, 1 (100101000101011111001)2 Binary Number Hexa decimal
In fact Mathematically speaking , every language, built on a declared
alphabet sets w.r.to the binary operation “ concatenation” is a Monoid.
20. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Consider the following phrases in english/தமிழ்
1. Apple is a fruit
2. Bharathiyar is a novalist
3. கதவை மூடவும்
4. Drop it !
5. முதுமவையிை் யாவையய
கிவடயாது
6. Wow! beautiful
7. 17+29 = 0
8. 2+7 > 5
9. நாை
் ச ாை்ைசதை்ைாம்
ச ாய்
1. True
2. False
3. Command
4. Command
5. False
6. Exclamation!
7. Relatively true
8. True
9. Paradox
truth value can be assigned are the needed “ Statement “
21. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
• “ Statements “ for which a definite truth value could be
assigned is called a proposition .
• which do not have any connecting words like and , or
, because, since… are called simple or atomic
statement other wise it is called compound statements.
• A statement formula is an expression involving one or
more statements connected by some logical
connectives.
22. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
• Depending on the language skills of persons same
statements can be stated with different words. But a
machine cannot understand all the possible equivalent
statements in general.
• Hence there is a need to symbolise the atomic
/simple propositions, using symbols /english alphabets like
p, q, r…. and the connecting words as Connectives so that
the entire language can be programmed easily .
• In view of this the logic we study is also called a symbolic
logic.
23. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
(i) 17 is a prime number and it is raining outside therefore
government teachers get the 5 % Dearness allowance.
Symbolise the atomic statements of the following sentences
p: 17 is a prime number
q: It is raining out side
r : Government teachers will get dearness allowance
(ii) 17 is a prime number and it is raining outside therefore
government teachers get the 5 % Dearness allowance.
24. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
24
Sl.No Name symbol English name
1
Negation
Not
2 Conjunction And
3 Disjunction Or
4 Conditional If...then...
5 Bi-conditional If and only if
Connectives
25. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Truth table
T F
F T
for negation
for conjunction
T T T
T F F
F T F
F F F
for disjunction
T T T
T F T
F T T
F F F
for conditional
T T T
T F F
F T T
F F T
for bi -conditional
T T T
T F F
F T F
F F T
for Exclusive OR
T T F
T F T
F T T
F F F
26. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Truth table contd…
for Nand
T T F
T F T
F T T
F F T
𝑝 ↑ 𝑞 = ¬(𝑝 ∧ 𝑞)
27. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Truth table contd… for Nor
T T F
T F F
F T F
F F T
𝑝 ↓ 𝑞 = ¬(𝑝 ∨ 𝑞)
28. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Construct the truth table for the statement
formula
(𝑝 ∨ 𝑞) ∧ (𝑝 ∨ ¬𝑞)
What ever be the truth value compositions of
‘p’ and ‘q’ the truth value of the given statement
formula is False.
29. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Construct the truth table for the statement
formula
((𝑝 ⟶ 𝑞) ∧ 𝑝) ⟶ 𝑞
What ever be the truth value compositions of
‘p’ and ‘q’ the truth value of the given statement
formula is True
30. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number
Symbolise the following statements
Let
p: 19 is a prime number
q: all the angles of a triangle are equal
(𝑖)¬𝑝 ∧ 𝑞 (𝑖𝑖)𝑝 ∧ ¬𝑞
(𝑖𝑖𝑖)𝑝 ∧ 𝑞 (𝑖𝑣)¬𝑝
31. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Classification of Statements
A statement is said to be a tautology if its
truth value is always T irrespective of the truth
values of its component statements. It is denoted
by 𝕋. It will be of the form , for some statement P
𝑃 ∨ ¬𝑃
A statement is said to be a contradiction if its
truth value is always F irrespective of the truth
values of its component statements. It is denoted
by F . It will be of the form , for some statement P
𝑃 ∧ ¬𝑃
A statement which is neither a contradiction nor a tautology is called a
contingency .
Note: In order to write the truth table for a given statement formulae
the number of rows needed are
where ’n’ is the number of atomic/simple statements.
2𝑛
32. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Equality of statements
Let P(p, q, r,), Q( p ,q, r ,…) be any two statements with same
simple statement variables . We say these two statements are
equivalent if the the truth values of these two statements are the same
irrespective of the truth values their components. In other words the
last column of P(p, q, r,…), Q( p ,q, r ,…) are the same. For example
𝑝 ⟶ 𝑞𝑎𝑛𝑑¬𝑝 ∨ 𝑞𝑎𝑟𝑒𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡
T T T T
T F F F
F T T T
F F T T
33. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Show the equivalence with and without the use of truth
table
Solution with out truth table
34. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Solution with truth table
p q r q—-> r p—->(q—->r) p^q (p^q) —->r
T T T T T T T
T T F F F T F
T F T T T F T
T F F T T F T
F T T T F
F T F F F
F F T T
F F F T
You may try it as an exercise
for other truth values
35. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Consequences of conditional statement
𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡𝑝 ⟶ 𝑞
𝐶𝑜𝑛𝑣𝑒𝑟𝑠𝑒𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡𝑞 ⟶ 𝑝
𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡¬𝑝 ⟶ ¬𝑞
𝐶𝑜𝑛𝑡𝑟𝑎𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡¬𝑞 ⟶ ¬𝑝
𝑝 ⟶ 𝑞𝑎𝑛𝑑¬𝑞 ⟶ ¬𝑝𝑎𝑟𝑒𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡
𝑝 ⟶ 𝑞 = ¬𝑞 ∨ 𝑝 = 𝑝 ∨ ¬𝑞
¬𝑞 ⟶ ¬𝑝 = ¬¬𝑝 ∨ ¬𝑞 = 𝑝 ∨ ¬𝑞
36. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Duality statements
37. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Dual of a statement
Let Q( p ,q, r ,…) be any statement or statement formula with
statement variables p, q, r ,… . The dual of Q is a statement say,
𝑄′
(𝑝, 𝑞, 𝑟) is statement obtained from Q by replacing
∨ 𝑏𝑦 ∧,∧ 𝑏𝑦 ∨, 𝑇𝑏𝑦𝐹, 𝐹𝑏𝑦𝑇
Duality law
𝑇ℎ𝑒𝑑𝑢𝑎𝑙𝑄′
(𝑝, 𝑞, 𝑟)𝑜𝑓𝑄(𝑝, 𝑞, 𝑟)𝑖𝑠
¬𝑄(¬𝑝, ¬𝑞, ¬𝑟)
38. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Considering the negation , conjunction and disjunction as
operators on the set of statement formulae , say S , we have
⟨𝑆,∧⟩𝑎𝑛𝑑⟨𝑆,∨⟩ are algebraic systems.
also these operators satisfies associative , commutative, distributive laws.
(𝑝 ∨ 𝑞) ∨ 𝑟 = 𝑝 ∨ (𝑞 ∨ 𝑟); 𝑝 ∧ (𝑞 ∧ 𝑟) = (𝑝 ∧ 𝑞) ∧ 𝑟
𝑝 ∨ 𝑞 = 𝑞 ∨ 𝑝; 𝑝 ∧ 𝑞 = 𝑞 ∧ 𝑝
𝑝 ∧ (𝑞 ∨ 𝑟) = (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟); 𝑝 ∨ (𝑞 ∧ 𝑟) = (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟
39. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
S.No Web Link For what
1
http://www-history.mcs.st-
and.ac.uk/Day_files/Year.html
To Know the mathematicians
of the day
2 http://www.wolframalpha.com/
To compute every thing in
mathematics
3
https://www.geeksforgeeks.org/p
roposition-logic/
for Mathematics logic
4 https://ocw.mit.edu/index.htm MIT Open course ware
40. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
e-mail: drtns2008@gmail.com
If you wish to contact me
you may do so through
Face book : Tirunelveli Nellaiappan Shanmugam
FB Group : Higher Secondary Mathematics Tamizh
Nadu
41. Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
நை
்
றி
பிரி
யைாம்
பிறியதாரிடத்
திை்
ந்தி ்ய ாம்