This document summarizes algebraic expressions used in regular expressions. It begins with an introduction to regular expressions and finite state automata. It then outlines four main laws of regular expression algebra:
1. Commutative and associative laws apply to operations like union and concatenation, allowing reordering and grouping of terms.
2. Identity elements (ε for concatenation) and annihilators (θ for concatenation) exist based on the operation.
3. Distributive laws allow distributing operations over each other.
4. Idempotent laws state that applying an operation to identical terms yields the original term.
Examples are provided to demonstrate applications of the laws in simplifying and proving equivalences of
This document provides an introduction to algebraic expressions and equations for middle school math. It includes sections on exponents, expressions, radicals, linear equations, and practice problems. The document contains information on the contents, fonts used, colors, icons, and editable presentation theme. It also includes examples and instructions for solving expressions and equations.
Pre-Calculus Lesson for High School_ Algebra and Equations by Slidesgo (2).pptxusmanghaniqasmi1
This document provides an overview of a pre-calculus lesson for high school students covering algebra and equations. It includes sections on algebraic concepts, equations and functions, and a review and practice section. The document contains tables of contents, slides with explanations of topics like variables, expressions, order of operations, and different types of equations. It also includes example problems for students to work through.
funciones de variable compleja, es una presentacion que puede ayudar a exponer el tema y asi ya no batallar tanto para hacer una nueva, tiene diapositivas a parte para que si se quiere agregar mas informacion o fotos, tiene conceptos basicos e imagenes.
tiene una decoracion acorde al tema, para darle formato.
5. Limit Fungsi yang menjadi Aljabar.pptxBanjarMasin4
This document discusses limits of algebraic functions in mathematics education. It begins with definitions of one-sided limits and general limits. It then covers theorems related to limits, including the Limit Theorems, Substitution Theorem, Squeeze Theorem, and the relationship between continuity and limits of functions. Examples are provided to demonstrate applying the theorems to calculate limits. The document provides foundations for understanding limits of functions through rigorous definitions and theorems.
Basic Calculus_Limits and Continuity.pptxJCNicolas2
This document provides an overview of limits and continuity in basic calculus for an 11th grade class. It begins with an essential question about the importance of calculus and explains that calculus is fundamental to fields like physics, engineering, economics, biology and computer science as it provides a foundation for understanding how things change over time. The document then discusses what limits and continuity are in mathematics, noting that limits express the concept of arbitrary closeness and allow understanding of calculus' differentiation and integration operations. It provides examples of illustrating limits using tables of values and graphs.
This document discusses algebraic expressions from a Venezuelan university. It defines algebraic expressions as combinations of letters and numbers using mathematical operations. It provides examples of algebraic expressions in geometry like the circumference of a circle. It also describes the elements of algebraic expressions like terms, coefficients, variables, and exponents. Finally, it explains different types of algebraic expressions and mathematical operations that can be performed on expressions like addition, subtraction, multiplication, and division.
This document provides an introduction to algebraic expressions and equations for middle school math. It includes sections on exponents, expressions, radicals, linear equations, and practice problems. The document contains information on the contents, fonts used, colors, icons, and editable presentation theme. It also includes examples and instructions for solving expressions and equations.
Pre-Calculus Lesson for High School_ Algebra and Equations by Slidesgo (2).pptxusmanghaniqasmi1
This document provides an overview of a pre-calculus lesson for high school students covering algebra and equations. It includes sections on algebraic concepts, equations and functions, and a review and practice section. The document contains tables of contents, slides with explanations of topics like variables, expressions, order of operations, and different types of equations. It also includes example problems for students to work through.
funciones de variable compleja, es una presentacion que puede ayudar a exponer el tema y asi ya no batallar tanto para hacer una nueva, tiene diapositivas a parte para que si se quiere agregar mas informacion o fotos, tiene conceptos basicos e imagenes.
tiene una decoracion acorde al tema, para darle formato.
5. Limit Fungsi yang menjadi Aljabar.pptxBanjarMasin4
This document discusses limits of algebraic functions in mathematics education. It begins with definitions of one-sided limits and general limits. It then covers theorems related to limits, including the Limit Theorems, Substitution Theorem, Squeeze Theorem, and the relationship between continuity and limits of functions. Examples are provided to demonstrate applying the theorems to calculate limits. The document provides foundations for understanding limits of functions through rigorous definitions and theorems.
Basic Calculus_Limits and Continuity.pptxJCNicolas2
This document provides an overview of limits and continuity in basic calculus for an 11th grade class. It begins with an essential question about the importance of calculus and explains that calculus is fundamental to fields like physics, engineering, economics, biology and computer science as it provides a foundation for understanding how things change over time. The document then discusses what limits and continuity are in mathematics, noting that limits express the concept of arbitrary closeness and allow understanding of calculus' differentiation and integration operations. It provides examples of illustrating limits using tables of values and graphs.
This document discusses algebraic expressions from a Venezuelan university. It defines algebraic expressions as combinations of letters and numbers using mathematical operations. It provides examples of algebraic expressions in geometry like the circumference of a circle. It also describes the elements of algebraic expressions like terms, coefficients, variables, and exponents. Finally, it explains different types of algebraic expressions and mathematical operations that can be performed on expressions like addition, subtraction, multiplication, and division.
Lecture: Regular Expressions and Regular LanguagesMarina Santini
This document provides an introduction to regular expressions and regular languages. It defines the key operations used in regular expressions: union, concatenation, and Kleene star. It explains how regular expressions can be converted into finite state automata and vice versa. Examples of regular expressions are provided. The document also defines regular languages as those languages that can be accepted by a deterministic finite automaton. It introduces the pumping lemma as a way to determine if a language is not regular. Finally, it includes some practical activities for readers to practice converting regular expressions to automata and writing regular expressions.
1. Complex numbers can be represented as a sum of a real part and an imaginary part, written as a + bi, where a is the real part and b is the imaginary part.
2. Operations like addition, subtraction, and multiplication on complex numbers follow the same rules as real numbers, with the exception that i2 = -1. Division of complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator.
3. Complex numbers can be represented graphically on an Argand diagram, with the real part along the x-axis and imaginary part along the y-axis. The modulus and argument can then be used to describe the magnitude and angle of the complex number.
This document discusses semantics and models in formal logic. It defines key terms like semantics, metalanguage, object language, logical symbols, non-logical symbols, interpretation, and models. Interpretation provides meaning for symbols and formulas, while models add factual information about how the interpreted symbols relate to the world. Truth and falsity of formulas depends on both interpretation and the state of the world. The document provides examples of assigning interpretations, constructing models that specify domains and extensions, and using models to evaluate formulas for truth. It concludes with practice problems assigning interpretations and models to evaluate formulas.
This 6-page document provides a guide to solving differential equations for electrical circuit analysis. It begins by dividing differential equations into homogeneous and non-homogeneous categories. Homogeneous equations have one solution, while non-homogeneous equations have two solutions: the complementary solution and particular solution. The complete solution is the sum of these. The document then discusses finding the complementary solution through the characteristic equation for 1st and 2nd order differential equations. It also covers finding the particular solution depending on whether the right-hand side is a polynomial, exponential, or combination of exponential and trigonometric terms. An example circuit problem is worked through to demonstrate the process.
This document discusses solving systems of linear equations with two or three variables. It explains that a system can have one solution, infinitely many solutions, or no solution. For two variables, the solutions are where lines intersect (one solution), coincide (infinitely many), or are parallel (no solution). For three variables, the solutions are where planes intersect (one point), lie on a line (infinitely many), or do not intersect (no solution). It demonstrates solving systems using substitution, elimination, and matrix methods, and discusses cases where a system has infinitely many or no solutions.
The document discusses linear equations, which involve variables raised to the first power. It provides examples of linear equations with one, two, and three variables. Linear equations can be used to solve real-world problems involving costs. The document also discusses representing linear equations graphically and solving systems of linear equations using various methods like substitution. Linear inequalities are also introduced, which involve inequality signs rather than equals signs. An example problem demonstrates solving a linear inequality for the variable.
The document discusses linear equations and their applications. It defines linear equations as equations where variables have a degree of one and do not involve products or roots of variables. Linear equations can be used to solve real-life problems involving costs and quantities. The document discusses different forms of linear equations with one, two, or three variables. It also discusses solving systems of linear equations using various methods like substitution. Graphs of linear equations are shown to be lines or points on a number line. Methods to solve and graph linear equations and inequalities are presented.
I am Frank Dennis. I love exploring new topics. Academic writing seemed an interesting option for me. After working for many years with mathhomeworksolver.com. I have assisted many students with their Assignments. I can proudly say, each student I have served is happy with the quality of the solution that I have provided. I have acquired my PhD. in Maths, Leeds University UK.
The document discusses the binomial theorem, which describes the pattern that emerges when a binomial is multiplied by itself multiple times. Specifically:
- When a binomial of the form (a + b) is raised to a power n, the terms follow the pattern an-kbk, where k goes from 0 to n.
- The coefficients of these terms form Pascal's triangle, where each number is the sum of the two above it.
- This allows one to expand complex expressions like (a + b)4 as a4 + 4a3b + 6a2b2 + 4ab3 + b4.
The binomial theorem thus provides a formulaic way to determine the terms and
The document provides definitions and explanations of key concepts in algebra including:
1. Types of numbers such as complex, rational, irrational, and integer numbers.
2. Properties of real numbers like commutative, associative, and distributive properties.
3. Exponents, radicals, logarithms, progressions, the binomial theorem, and word problems.
This document provides information and examples about algebra rules and concepts including:
- Assigning variables to unknown quantities, such as assigning 'm' to the person who weighs the least.
- Identifying like terms in algebraic expressions as terms that contain the same variables raised to the same power, even if the coefficients are different.
- Providing formulas for perimeter and area of squares and triangles.
- Examples of translating word problems into algebraic expressions and equations, and simplifying expressions by combining like terms.
- A review of topics to be covered on Test #1, including simplifying expressions, solving simple one- and two-step equations, and translating sentences into algebraic representations.
The document presents a multi-step problem to determine the area bounded by a circle, the x-axis, y-axis, and the line y=12. It provides the equation of the circle and guides the reader through simplifying systems of equations to find values for variables in the circle equation. These are used to graph the circle and identify the bounded area as a rectangle minus one quarter of the circle area, calculated as 66.9027 square units.
1. The document discusses advanced counting methods including the principle of inclusion-exclusion and solving recurrence relations.
2. It provides examples of using inclusion-exclusion to find the number of elements in a union of sets that satisfy certain properties.
3. It also discusses solving homogeneous and non-homogeneous linear recurrence relations of first and second order using characteristic roots and the method of undetermined coefficients.
The math tutoring session covered solving equations using order of operations and variables, calculating perimeter, area, and circumference, and word problems involving rates and time. Key objectives were for the student to learn order of operations, solve equations, and review calculating perimeter, area, and circumference. Sample problems were provided to demonstrate solving techniques.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
The document provides an introduction to physics concepts and topics covered in the Mexican upper secondary education program COMIPEMS. It includes sections on motion, forces, interactions of matter, and the internal structure of matter. Key concepts defined include physics, physical phenomena, and basic mathematical symbols used. Conversion of units between the metric, CGS, and English systems are also presented.
Lecture: Regular Expressions and Regular LanguagesMarina Santini
This document provides an introduction to regular expressions and regular languages. It defines the key operations used in regular expressions: union, concatenation, and Kleene star. It explains how regular expressions can be converted into finite state automata and vice versa. Examples of regular expressions are provided. The document also defines regular languages as those languages that can be accepted by a deterministic finite automaton. It introduces the pumping lemma as a way to determine if a language is not regular. Finally, it includes some practical activities for readers to practice converting regular expressions to automata and writing regular expressions.
1. Complex numbers can be represented as a sum of a real part and an imaginary part, written as a + bi, where a is the real part and b is the imaginary part.
2. Operations like addition, subtraction, and multiplication on complex numbers follow the same rules as real numbers, with the exception that i2 = -1. Division of complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator.
3. Complex numbers can be represented graphically on an Argand diagram, with the real part along the x-axis and imaginary part along the y-axis. The modulus and argument can then be used to describe the magnitude and angle of the complex number.
This document discusses semantics and models in formal logic. It defines key terms like semantics, metalanguage, object language, logical symbols, non-logical symbols, interpretation, and models. Interpretation provides meaning for symbols and formulas, while models add factual information about how the interpreted symbols relate to the world. Truth and falsity of formulas depends on both interpretation and the state of the world. The document provides examples of assigning interpretations, constructing models that specify domains and extensions, and using models to evaluate formulas for truth. It concludes with practice problems assigning interpretations and models to evaluate formulas.
This 6-page document provides a guide to solving differential equations for electrical circuit analysis. It begins by dividing differential equations into homogeneous and non-homogeneous categories. Homogeneous equations have one solution, while non-homogeneous equations have two solutions: the complementary solution and particular solution. The complete solution is the sum of these. The document then discusses finding the complementary solution through the characteristic equation for 1st and 2nd order differential equations. It also covers finding the particular solution depending on whether the right-hand side is a polynomial, exponential, or combination of exponential and trigonometric terms. An example circuit problem is worked through to demonstrate the process.
This document discusses solving systems of linear equations with two or three variables. It explains that a system can have one solution, infinitely many solutions, or no solution. For two variables, the solutions are where lines intersect (one solution), coincide (infinitely many), or are parallel (no solution). For three variables, the solutions are where planes intersect (one point), lie on a line (infinitely many), or do not intersect (no solution). It demonstrates solving systems using substitution, elimination, and matrix methods, and discusses cases where a system has infinitely many or no solutions.
The document discusses linear equations, which involve variables raised to the first power. It provides examples of linear equations with one, two, and three variables. Linear equations can be used to solve real-world problems involving costs. The document also discusses representing linear equations graphically and solving systems of linear equations using various methods like substitution. Linear inequalities are also introduced, which involve inequality signs rather than equals signs. An example problem demonstrates solving a linear inequality for the variable.
The document discusses linear equations and their applications. It defines linear equations as equations where variables have a degree of one and do not involve products or roots of variables. Linear equations can be used to solve real-life problems involving costs and quantities. The document discusses different forms of linear equations with one, two, or three variables. It also discusses solving systems of linear equations using various methods like substitution. Graphs of linear equations are shown to be lines or points on a number line. Methods to solve and graph linear equations and inequalities are presented.
I am Frank Dennis. I love exploring new topics. Academic writing seemed an interesting option for me. After working for many years with mathhomeworksolver.com. I have assisted many students with their Assignments. I can proudly say, each student I have served is happy with the quality of the solution that I have provided. I have acquired my PhD. in Maths, Leeds University UK.
The document discusses the binomial theorem, which describes the pattern that emerges when a binomial is multiplied by itself multiple times. Specifically:
- When a binomial of the form (a + b) is raised to a power n, the terms follow the pattern an-kbk, where k goes from 0 to n.
- The coefficients of these terms form Pascal's triangle, where each number is the sum of the two above it.
- This allows one to expand complex expressions like (a + b)4 as a4 + 4a3b + 6a2b2 + 4ab3 + b4.
The binomial theorem thus provides a formulaic way to determine the terms and
The document provides definitions and explanations of key concepts in algebra including:
1. Types of numbers such as complex, rational, irrational, and integer numbers.
2. Properties of real numbers like commutative, associative, and distributive properties.
3. Exponents, radicals, logarithms, progressions, the binomial theorem, and word problems.
This document provides information and examples about algebra rules and concepts including:
- Assigning variables to unknown quantities, such as assigning 'm' to the person who weighs the least.
- Identifying like terms in algebraic expressions as terms that contain the same variables raised to the same power, even if the coefficients are different.
- Providing formulas for perimeter and area of squares and triangles.
- Examples of translating word problems into algebraic expressions and equations, and simplifying expressions by combining like terms.
- A review of topics to be covered on Test #1, including simplifying expressions, solving simple one- and two-step equations, and translating sentences into algebraic representations.
The document presents a multi-step problem to determine the area bounded by a circle, the x-axis, y-axis, and the line y=12. It provides the equation of the circle and guides the reader through simplifying systems of equations to find values for variables in the circle equation. These are used to graph the circle and identify the bounded area as a rectangle minus one quarter of the circle area, calculated as 66.9027 square units.
1. The document discusses advanced counting methods including the principle of inclusion-exclusion and solving recurrence relations.
2. It provides examples of using inclusion-exclusion to find the number of elements in a union of sets that satisfy certain properties.
3. It also discusses solving homogeneous and non-homogeneous linear recurrence relations of first and second order using characteristic roots and the method of undetermined coefficients.
The math tutoring session covered solving equations using order of operations and variables, calculating perimeter, area, and circumference, and word problems involving rates and time. Key objectives were for the student to learn order of operations, solve equations, and review calculating perimeter, area, and circumference. Sample problems were provided to demonstrate solving techniques.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
The document provides an introduction to physics concepts and topics covered in the Mexican upper secondary education program COMIPEMS. It includes sections on motion, forces, interactions of matter, and the internal structure of matter. Key concepts defined include physics, physical phenomena, and basic mathematical symbols used. Conversion of units between the metric, CGS, and English systems are also presented.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
artificial intelligence and data science contents.pptxGauravCar
What is artificial intelligence? Artificial intelligence is the ability of a computer or computer-controlled robot to perform tasks that are commonly associated with the intellectual processes characteristic of humans, such as the ability to reason.
› ...
Artificial intelligence (AI) | Definitio
2. Anggota Kelompok
1. Azzah Shaffiyah (2065061001)
2. Beltra Saura Rahmadan (2015061029)
3. Muhamad Ferdian Hidayat (2015061030)
4. Muhammad Rizky Rifaldi (2055061002)
5. Rahmad Romadhoni (2015061033)
6. Zaki Taufiqurrachman (2015061034)
3. • Sebuah bahasa dinyatakan
memiliki ekspresi regular jika
terdapat Finite State Automata
(FSA) yang dapat menerimanya.
Ekspresi Reguler
• Bahasa-bahasa yang diterima
oleh FSA bisa dinyatakan secara
sederhana dengan ekspresi
regular (regular expression).
4. bisa tidak muncul,
bisa juga muncul
berhingga kali
ER : ab*cc, 010*, a*d
berarti minimal
muncul satu kali
ER : a⁺d
union
ER : a * ∪b * , (a ∪b) *,01 * + 0
konkatenansi
Biasanya tanpa ditulis
titiknya, misal ab, berarti
sama dengan a.b
Notasi Reguler
* + + U
atau .(titik)
6. 01. Sifat Komutatif
dan Asosiatif
• Sifat Komutatif
Dapat membalik urutan operand-operand nya
dan tetap memperoleh hasil akhir yang sama.
Memungkinkan kita untuk mengelompokkan
operand nya Ketika operator dikenakan dua kali.
• Sifat Asosiatif
7. Sifat Komutatif & Asosiatif
yang berlaku pada ER
1) A + B = B + A
Hukum ini (hukum komutatif untuk gabungan) menyatakan
bahwa kita dapat melakukan gabungan dua bahasa
tersebut, baik dengan urutan seperti di sebelah kiri ‘=‘
maupun seperti di sebelah kanan ‘=‘
8. 2) (A + B ) + C = A + ( B + C )
Hukum ini (hukum asosiatif untuk gabungan) menyatakan bahwa kita dapat
melakukan gabungan pada tiga bahasa, baik dengan mengambil gabungan dua
bahasa pertama terlebih dahulu maupun dengan mengambil gabungan dau
bahasa terakhir.
3) ( AB ) C = A ( BC )
Hukum ini ( hukum asosiatif untuk penyambungan/concatenation) menyatakan
bahwa kita dapat merenteng tuga bahasa dengan menyambung dua bahasa
pertama terlebih dahulu atau dua bahasa terakhir terlebih dahulu.
9. Hukum : AB = BA tidak berlaku dalam ekspresi regular
Contoh : ekspresi regular 01 dan 10
Ekspresi tersebut berturut-turut melambangkan bahasa {01} dan {10}.
Ekspresi 0 untuk A dan 1 untuk B tidak dapat disubstitusi.
Karena bahasanya berbeda, aka hukum AB = BA tidak berlaku.
10. 02. Identidas dan Anihilator
a) Identitas suatu operator
Nilai yang sedemikian sehingga Ketika dikenakan pada identitas dan suatu
nilai lain, maka hasilnya nilai lain lagi.
Contoh :
0 adalah identitas untuk penjumlahan, karena
0 + ꭓ = ꭓ + 0 = ꭓ
1 adalah identitas untuk perkalian, karena
1 x ꭓ = ꭓ x 1 = ꭓ
11. b) Anihilator untuk suatu operator
Nilai yang sedemikian sehingga Ketika operator tersebut dikenakan pada
annihilator dan suatu nilai lain, hasilnya adalah annihilator.
Contoh :
0 adalah annihilator untuk perkalian, karena
0 x ꭓ = ꭓ x 0 = 0
12. Hukum identitas dan annihilator
yang berlaku pada ER
1) θ + L = L + θ = L
Hukum ini menegaskan bahwa θ adalah identitas untuk
operasi gabungan.
2) ϵ L = L ϵ = L
Hukum ini menegaskan bahwa ϵ adalah identitas untuk
operasi concatenation.
3) θ L = L θ = θ
Hukum ini menegaskan bahwa θ adalah annihilator untuk
operasi concatenation.
13. 03. Hukum Distributif
Hukum distributive melibatkan dua operator, dan menyatakan
bahwa salah satu operator dapat dipaksa untuk dikenakan pada
tiap-tiap argument operator lain secara individual.
Contoh :
Hukum distributive perkalian atas penjumlahan
ꭓ x ( y + z ) = ( ꭓ x y ) + ( ꭓ x z )
14. Hukum distributive yang
berlaku pada ER
1) A ( M + N ) = AM + AN
Hukum ini adalah hukum distributive kiri
concatenation terhadap gabungan (union).
2) ( M + N ) A = MA + NA
Hukum ini adalah hukum distributive kanan
concatenation terhadap gabungan (union).
Gabungan dan irisan adalah contoh sederhana
operator idempotent.
15. 04. Hukum Idempoten
Suatu Operator dikatakan idempotent jika hasil
penerapannya pada dua nilai yang sama sebagai
argument adalah nilai itu sendiri.
Operator aritmatika biasa tidak bersifat idempotent ;
ꭓ + ꭓ ≠ ꭓ dan ꭓ x ꭓ ≠ ꭓ
Gabungan dan irisan adalah contoh sederhana
operator idempotent.
16. Hukum idempotent yang
berlaku pada ER
1) L + L = L
Hukum ini (hukum idempoten untuk gabungan)
menyatakan bahwa jika kita mengambil
gabungan dari dua ekspresi yang identic, kita
dapat mengganti keduanya dengan satu Salinan
ekspresi tersebut.
17. Contoh Soal
1. Diberikan ekspresi regular 0 + 01*.
Penyelesaian :
Ekspresi tersebut dapat disederhanakan menggunakan hukum-hukum
aljabar dalam ekspresi regular:
0 + 01*
= 0ε + 01* dari (2b)
= 0(ε + 1*) dari (3a), distributif kiri
= 01* karena ε + R = R
18. Contoh Soal
2. (L + M )* = (L*M*)*
Penyelesaian :
Untuk menunjukkan kesamaan tersebut, ganti variabel L dan M berturut-turut
dengan symbol a dan b, sehingga diperoleh ekspresi regular (a+b)* dan (a*b*)*.
Kedua ekspresi regular tersebut menyatakan bahasa dengan semua string dari a
dan b.
Dengan demikian, kesamaan (L + M )* = (L*M*)* benar.
19. Contoh Soal
3. L* = L*L*
Penyelesaian :
Untuk menunjukkan kesamaan tersebut, ganti
variabel L dengan simbol a, sehingga diperoleh
ekspresi regular a* dan a*a*.
Kedua ekspresi regular tersebut menyatakan
bahasa dengan semua string dari a.
Dengan demikian, kesamaan L* =L*L* benar.
20. Contoh Soal
4. L + ML = (L + M)L
Penyelesaian :
Untuk menunjukkan kesamaan tersebut, ganti
variabel L dan M berturut-turut dengan simbol a
dan b, sehingga diperoleh ekspresi regular a+ba
dan (a+b)a.
Kedua ekspresi regular tersebut menyatakan
Bahasa yang berbeda.
Untuk menunjukkan hal tersebut, pilih aa dalam
bahasa dari ekspresi regular (a+b)a, tapi tidak
dalam bahasa dari ekspresi regular a+ba.
Dengan demikian, kesamaan L + ML = (L + M)L
salah
25. Mercury
It’s the smallest planet
of them all
Venus
Venus is the second
planet from the Sun
Jupiter
Jupiter is the biggest
planet of them all
Saturn
It’s composed of
hydrogen and helium
Mars
Mars is actually a very
cold place
Neptune
It’s the farthest planet
from the Sun
Algebraic expressions
26. Algebraic operations
Venus
Venus is the second
planet from the Sun
Jupiter
Jupiter is the biggest
planet of them all
Mars
Despite being red,
Mars is a cold place
Saturn
Saturn is a gas giant
and has several rings
27. A picture always
reinforces the
concept
Images reveal large amounts of data, so
remember: use an image instead of a long text.
Your audience will appreciate it
28. This is a map
Latin America
Venus has a beautiful name
and is the second planet
from the Sun
Europe
Despite being red, Mars is
actually a cold place. It’s full
of iron oxide dust
29. Steps to tackle equations
01
Read
problem
Venus is the
second planet
from the Sun
02
Identify
unknown
Jupiter is the
biggest planet of
them all
03
Approach
equation
Despite being
red, Mars is a
cold place
04
Solve
equation
Saturn is a gas
giant and has
several rings
05
Answer
question
Mercury is the
smallest planet of
them all
30. X - 5 = 3
Example of an equation
Unknown
Jupiter is the biggest
planet of them all
First member
Saturn is composed of
hydrogen and helium
Second member
Neptune is very far away
from Earth
31. Activities
● A number increased by 8 equals 22. What number is it? --
● Maria's age increased by 7 equals 15. How old is Maria? --
● Double a number equals 46. Find that number --
● If half of Marta's age is equal to 10, how old is Marta? --
● The triple of a number decreased by 1 is equal to 20. Find that number --
32. Do these exercises
Solve: Result
7 x 4 --
6 x 8 --
3 x 9 --
5 x 5 --
4 x 8 --
Solve: Result
3 x 6 --
2 x 8 --
9 x 2 --
7 x 9 --
2 x 6 --
33. Practical exercise
Exercise A
Venus has a beautiful
name and is the second
planet from the Sun
Exercise B
Despite being red, Mars
is actually a cold place.
It’s full of iron oxide dust
Result
--
Result
--
34. Interpretation exercise
Interpret a multiplication equation as a comparison, for example:
35 = 5 × 7 35 is 5 times as
many as 7
7 times as many
as 5
Comparison A
Despite being red, Mars is
a very cold place
Comparison B
Saturn is a gas giant and
has several rings
Interpret
Neptune is the farthest
planet from the Sun
35. Multiples
Observe and match:
Multiples of 5
Multiples of 9
Multiples of 6
Multiples of 8
Multiples of 7
Multiples of 4
12, 18, 24, 30
16, 24, 32, 40
15, 20, 25, 30
18, 27, 36,45
14, 28, 35, 42
28, 36, 40, 44
36. Important concepts
Factors
Jupiter is the biggest
planet of them all
Hexagons
Saturn is a gas giant
and has several rings
Equality
Neptune is very far
away from us
Logarithm
Ceres is located in
the asteroid belt
Binomial
Mercury is the
smallest planet
Cube
Venus is the second
planet from the Sun
Digit
Mars is actually a
very cold place
Exponent
Earth is the planet
where we live on
38. Tablet
app
You can replace the image on
the screen with your own work.
Just right-click on it and select
“Replace image”
39. Mobile
app
You can replace the image on the
screen with your own work. Just
right-click on it and select
“Replace image”
40. Susan Bones Timmy Jimmy
You can talk a bit about this
person here
You can talk a bit about this
person here
Our team
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