Ho class 1 handouts
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The document defines and provides examples of different types of sets and numbers, including: 1) Sets can contain distinct objects and are defined using roster notation or set-builder notation. A set is a subset of another if it contains the same elements. 2) The union of two sets contains elements that are in either set, while the intersection contains elements in both sets. 3) Natural numbers are closed under addition but not subtraction. Integers are closed under both. Rational numbers can be expressed as fractions and are closed under all operations. Irrational numbers have non-terminating, non-repeating decimals. 4) The real numbers contain rational and irrational numbers and can be represented on a number
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The document involves calculating inverse functions of various compositions of functions f(x) and g(x). It provides the definitions of f(x) and g(x) and calculates:
1) The inverse functions f^-1(x) and g^-1(x)
2) The compositions (f o g)(x), (g o f)(x), and their inverse functions
3) The inverse functions of (f o g)^-1(x) and (g o f)^-1(x)
4) Confirms the relationship that (f o g)^-1(x) = (g^-1 o f^-1)(x) and (g o f)^-1
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Ans Hw Lesson 8
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الحل العام للمعادلة المثلثية
The document discusses the general solutions to three types of cubic equations:
1) For equations of the form α = βγ, the general solution is α = nβ + (-1)^n β.
2) For equations of the form α = βγ^2, the general solution is 2α = nβ ± β.
3) For equations of the form α = βγ^3, the general solution is α = nβ + β.
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Holiday assignment
This document contains 10 math problems involving factoring polynomials, solving inequalities, and expressing fractions in partial fraction decomposition form. It asks the reader to find the values of coefficients a and b in two polynomials based on given factors and remainders, determine the sets of values for which certain inequalities are true, and express fractions as sums of partial fractions.
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Invers fungsi
The document involves calculating inverse functions of various compositions of functions f(x) and g(x). It provides the definitions of f(x) and g(x) and calculates:
1) The inverse functions f^-1(x) and g^-1(x)
2) The compositions (f o g)(x), (g o f)(x), and their inverse functions
3) The inverse functions of (f o g)^-1(x) and (g o f)^-1(x)
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The document contains homework directions to evaluate functions f(x), g(x), and h(x) for various values of x. f(x) is defined as -x^2 - 9, g(x) is defined as 3x - 9, and h(x) is defined as 2x^2 + 4x - 9. There are 12 problems that apply these functions, such as finding f(-4), h(-3), g(1/2), and g(h(2/3)).
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The document contains a series of math problems involving equations with variables and their manipulation through factorization, solving for values, and proving statements. The problems cover topics like quadratic equations, factorization of polynomials, and manipulation of radical and fractional expressions.
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The document discusses the general solutions to three types of cubic equations:
1) For equations of the form α = βγ, the general solution is α = nβ + (-1)^n β.
2) For equations of the form α = βγ^2, the general solution is 2α = nβ ± β.
3) For equations of the form α = βγ^3, the general solution is α = nβ + β.
Examples are provided to demonstrate applying the general solutions to specific cubic equations.
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This document contains 10 math problems involving factoring polynomials, solving inequalities, and expressing fractions in partial fraction decomposition form. It asks the reader to find the values of coefficients a and b in two polynomials based on given factors and remainders, determine the sets of values for which certain inequalities are true, and express fractions as sums of partial fractions.
Definationofset
This document defines several key concepts in set theory, including:
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3) There are different types of sets such as finite sets with a countable number of elements, infinite sets, singleton/unit sets with one element, and the empty set with no elements.
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The document provides information about an online mathematics class for 8th grade students. It discusses the topic of sets and power sets. It defines the formula for calculating the size of a power set as P=2^n, where P is the power set and n is the number of elements in the original set. As an example, it calculates the power set of the set {a,b,c,d} as having 16 sets. It provides the list of all 16 possible subsets. It then assigns mathematics homework from the class involving problems on sets and power sets and provides a link to a YouTube video explaining the concepts.
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Link download full: Solution Manual for A Survey of Mathematics with Applications 8th Edition by Allen R. Angel, Christine D. Abbott and Dennis C. Runde
https://digitalcontentmarket.org/download/solution-manual-for-a-survey-of-mathematics-with-applications-8th-edition-by-angel-abbott-and-runde/
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2. G consists of elements of the form (a,b,c,d,e) where a,b,c,d,e belong to {0,1,2,...,N-1} under a binary operation defined by (a,b,c,d,e)*(f,g,h,i,k)=(x,y,z,w,l) where the values are computed modulo N.
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This document provides instructions for transposing formulae by changing the subject to different variables. It includes examples of transposing formulae where the subject is changed to w, t, r, n, and t, as well as transposing formulae where the subject is changed to a given variable in square brackets. The examples are shown step-by-step with the work clearly shown at each stage of transposing the formulae.
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The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, division, and composition. It provides examples of evaluating each type of operation on functions by first evaluating the individual functions at a given value or variable and then performing the indicated operation on the results. Composition involves evaluating the inner function first and substituting its result into the outer function.
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- Two sets are equal if they have exactly the same elements. A set A is a subset of set B if every element of A is also an element of B.
- The power set of a set A is the collection of all subsets of A, including the empty set and A itself. It is denoted by P(A).
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A survey was conducted of 1,000 people who like to travel in Phetchabun. The following information was found:
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- 560 people like to travel by motorcycle
- 100 people like to travel by bus and car
- 50 people like to travel by bus and motorcycle
- 40 people like to travel by car and motorcycle
- 30 people like to travel by bus, car, and motorcycle
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De Morgan's Laws are transformational Rules for 2 Sets
1) Complement of the Union Equals the Intersection of the Complements
not (A or B) = not A and not B
2) Complement of the Intersection Equals the Union of the Complements
not (A and B) = not A or not B
Take 2 Sets A and B
Union = A U B ← Everything in A or B
Intersection = A ∩ B ← Everything in A and B
U = Universal Set (All possible elements in your defined universe)
Complement = A’ Everything not in A, but in the Universal Set
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The document defines 9 functions:
1. g(x) is defined for all real numbers except x=4.
2. F(x) is defined for all real numbers except x=0.
3. h(x) is defined for all real numbers.
4. G(x) is defined for all real numbers except x=0.
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2) Elements are the individual objects that make up a set. Sets can be defined using a tabular method by listing elements or a set-builder method using properties of the elements.
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Comp decomp worked
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This document discusses set operations and identities. It defines operations like union, intersection, complement, difference, and cardinality. It presents examples of calculating unions, intersections, complements, and differences of sets. It also covers set identities like commutative, associative, distributive, De Morgan's laws, and absorption laws. Methods for proving identities like subset proofs and membership tables are described. An example proof of the second De Morgan's law is provided using subset notation.
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Solution manual for a survey of mathematics with applications 8th edition by ...
Link download full: Solution Manual for A Survey of Mathematics with Applications 8th Edition by Allen R. Angel, Christine D. Abbott and Dennis C. Runde
https://digitalcontentmarket.org/download/solution-manual-for-a-survey-of-mathematics-with-applications-8th-edition-by-angel-abbott-and-runde/
a survey of mathematics with applications 8th edition solution manual
a survey of mathematics with applications pdf solution manual
solution manual a survey of mathematics with applications 8th edition pdf
a survey of mathematics with applications download solution manual
solution manual a survey of mathematics with applications free pdf
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1. The document defines a non-abelian group G of order N=PQRSp+1, where P, Q, R, S are distinct positive integers greater than or equal to 2.
2. G consists of elements of the form (a,b,c,d,e) where a,b,c,d,e belong to {0,1,2,...,N-1} under a binary operation defined by (a,b,c,d,e)*(f,g,h,i,k)=(x,y,z,w,l) where the values are computed modulo N.
3. The document states that multiplying an element of G by itself
Specific Finite Groups(General)
This document provides an introduction and overview of Sylow's theorem regarding the construction of finite groups with specific numbers of Sylow p-subgroups. It begins with prerequisites and definitions, then presents three theorems:
Theorem 1 proves the existence of a group with qe Sylow p-subgroups for any e in a set E. Corollary 1 extends this to allow constructing groups with qem Sylow p-subgroups for any m. Theorem 2 addresses the special case of 2-subgroups, showing there exists a group with n Sylow 2-subgroups for any odd positive integer n. The document establishes notation and provides proofs of lemmas supporting each theorem. It aims to provide intuition on constructing groups to
Change of subject
This document provides instructions for transposing formulae by changing the subject to different variables. It includes examples of transposing formulae where the subject is changed to w, t, r, n, and t, as well as transposing formulae where the subject is changed to a given variable in square brackets. The examples are shown step-by-step with the work clearly shown at each stage of transposing the formulae.
Statistical Inference Using Stochastic Gradient Descent
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Many areas of machine learning and data mining focus on point estimates of key parameters. In transportation, however, the inherent variance, and, critically, the need to understand the limits of that variance and the impact it may have, have long been understood to be important. Indeed, variance and other risk measures that capture the cost of the spread around the mean, are critical factors in understanding how people act. Thus they are critical for prediction, as well as for purposes of long term planning, where controlling risk may be equally important to controlling the mean (the point estimate).
There has been tremendous progress on large scale optimization techniques to enable the solution of large scale machine learning and data analytics problems. Stochastic Gradient Descent and its variants is probably the most-used large-scale optimization technique for learning. This has not yet seen an impact on the problem of statistical inference — namely, obtaining distributional information that might allow us to control the variance and hence the risk of certain solutions.Operation on functions
The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, division, and composition. It provides examples of evaluating each type of operation on functions by first evaluating the individual functions at a given value or variable and then performing the indicated operation on the results. Composition involves evaluating the inner function first and substituting its result into the outer function.
SETS
- A set is a well-defined collection of objects. The empty set is a set with no elements. A set is finite if it has a definite number of elements, otherwise it is infinite.
- Two sets are equal if they have exactly the same elements. A set A is a subset of set B if every element of A is also an element of B.
- The power set of a set A is the collection of all subsets of A, including the empty set and A itself. It is denoted by P(A).
Sets
A survey was conducted of 1,000 people who like to travel in Phetchabun. The following information was found:
- 320 people like to travel by bus
- 240 people like to travel by car
- 560 people like to travel by motorcycle
- 100 people like to travel by bus and car
- 50 people like to travel by bus and motorcycle
- 40 people like to travel by car and motorcycle
- 30 people like to travel by bus, car, and motorcycle
The number of people who like to travel by at least one mode is the total of all the categories listed above. Therefore, the number of people who like to travel by at least one way is 320 + 240 + 560 +
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JEE Mathematics / Lakshmikanta Satapathy / Questions and Answers on Conditional Probability which includes Problems on two dice and selecting questions from a question bank
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The document is a quiz on order of operations and the distributive property with 4 multiple choice questions. It tests evaluating expressions using order of operations, identifying the order of operations in expressions, distributing terms in expressions, and simplifying expressions using the distributive property.
How to Prove and Apply De Morgan's Laws
De Morgan's Laws Proof and real world application.
De Morgan's Laws are transformational Rules for 2 Sets
1) Complement of the Union Equals the Intersection of the Complements
not (A or B) = not A and not B
2) Complement of the Intersection Equals the Union of the Complements
not (A and B) = not A or not B
Take 2 Sets A and B
Union = A U B ← Everything in A or B
Intersection = A ∩ B ← Everything in A and B
U = Universal Set (All possible elements in your defined universe)
Complement = A’ Everything not in A, but in the Universal Set
Natalia rosales duban erazo
The document defines 9 functions:
1. g(x) is defined for all real numbers except x=4.
2. F(x) is defined for all real numbers except x=0.
3. h(x) is defined for all real numbers.
4. G(x) is defined for all real numbers except x=0.
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Solution manual for a survey of mathematics with applications 8th edition by ...
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1. The document provides information about a teacher's schedule and the standards and objectives for a lesson on sets and Venn diagrams.
2. It gives examples of using Venn diagrams to represent sets and set operations like complement, intersection, union, and difference.
3. In a word problem, students are asked to identify complements, intersections, unions and differences of sets based on Venn diagram representations.
Sets
A Brief introduction to set and sets types. in very simple method that can every one enjoy during study.
#inshallah.
Operations on Sets
This will help you in illustrating operations on sets using Venn Diagram. Also, how to find the intersection, union, complement and difference of two sets.
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mathematical sets.pdf
The document provides information about sets including:
1) Sets can be finite, infinite, empty, or singleton and are represented using curly brackets. Common set operations are union, intersection, and complement.
2) A set's cardinality refers to the number of elements it contains. Subsets are sets contained within other sets.
3) Examples are given of representing sets using roster and set-builder methods and performing set operations like union, intersection, and complement on sample sets.
4) Basic properties of sets like idempotent, commutative, and associative laws for simplifying set expressions are outlined.
Set theory
Set theory is the study of sets, which are collections of objects known as elements. A set can be represented through description using words, set notation using curly brackets, or Venn diagrams. Common set operations include union, intersection, subsets, complements, and the empty set. For two sets to be equal, they must contain all the same elements. The intersection of sets contains elements that are common to both sets, while the union includes all elements in either set.
Edu presentation 1
Set theory is the study of sets, which are collections of objects known as elements. A set can be represented through description using words, set notation using curly brackets, or Venn diagrams. Common set operations include union, intersection, subsets, complements, and the empty set. For two sets to be equal, they must contain all the same elements. The intersection of sets contains elements that are common to both sets, while the union includes all elements in either set.
Universal Set and Subset.pptx
The document defines universal sets, subsets, and proper subsets. A universal set contains all elements being considered, while a subset contains elements that are also in the universal set. A proper subset is a subset where the universal set has at least one element not in the subset. The number of subsets of a set is equal to 2 to the power of the number of elements in the set.
Set Concepts
1) Set theory helps organize things into groups and understand logic. Key contributors include Georg Cantor, John Venn, George Boole, and Augustus DeMorgan.
2) A set is a collection of elements. A subset contains only elements that are also in another set. The cardinality of a set refers to the number of elements it contains.
3) Venn diagrams show relationships between sets using overlapping circles to represent their common elements.
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
Math lesson for Grade 7 students
Topic: Basic Concept of Sets
1Q LP 3.4 Venn Diagram & Set Operation (Complement and DIfference).ppsx
This document discusses set operations including union, intersection, complement, and difference of sets. It provides examples of determining the elements that belong to each set operation using Venn diagrams and set notation. It also discusses determining the cardinality or number of elements in each set operation. The learning targets are to recall and differentiate between various set operations and determine the relevant elements and regions.
Set concepts
The document defines key concepts in set theory including:
1. A set is a well-defined collection of distinct objects called elements. Georg Cantor is credited with creating set theory.
2. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties the elements share.
3. Operations on sets include intersection, union, difference, and complement. Intersection is the set of common elements, union is all elements in either set, difference is elements only in the first set, and complement is elements not in the set.
Set relationships
This document discusses sets and set operations including:
- Defining a universal set, Venn diagrams, equal sets, subsets, and the null set { }.
- Set operations such as union, intersection, disjoint sets, set difference, and complements.
- Examples are provided to demonstrate each concept such as finding subsets, unions, intersections, differences and complements of given sets.
- A word problem involving sets of students and modes of transportation is presented to demonstrate problem solving using sets.
Sets in Maths (Complete Topic)
Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
CMSC 56 | Lecture 6: Sets & Set Operations
A set is an unordered collection of objects called elements. A set contains its elements. Sets can be finite or infinite. Common ways to describe a set include listing elements between curly braces or using a set builder notation. The cardinality of a set refers to the number of elements it contains. Power sets contain all possible subsets of a given set. Cartesian products combine elements of sets into ordered pairs. Common set operations include union, intersection, difference, and testing if sets are disjoint.
TASK 1.docx
This document contains a 10-day unit plan for a 7th grade mathematics unit on sets and the real number system. It includes:
- Daily activities covering key concepts like sets, subsets, Venn diagrams, operations on integers, and using diagrams to solve problems
- Examples of terms, statements, and multiple choice questions to reinforce lessons
- Exercises identifying subsets, set operations like unions and intersections, and set differences
- A Venn diagram activity asking students to determine information like the number of students in particular clubs or no clubs
- A roster form for one of the sets in the Venn diagram
GRADE 7 MATHEMATICS - QUARTER 1 (WEEK 1).pptx
The document discusses sets and their properties. It defines sets as collections of objects and their elements. Finite sets can be counted, while infinite sets cannot. Two sets are equal if they contain the same elements, and equivalent if there is a one-to-one correspondence between elements. Sets can be defined using roster or set-builder notation.
Set theory
The document provides notes on set theory concepts including:
1. Sets are well-defined collections of objects called elements or members. Examples of sets include collections of boys in a class or numbers within a specified range.
2. For a collection to be considered a set, the objects must be well-defined so it is clear which objects are members.
3. There are different ways to represent sets including roster form listing elements, statement form describing characteristics of elements, and rule form using logical statements.
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Ho class 1 handouts
- 1. Set: A well-defined collection of distinct objects. 1
- 2. d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9, pw, sw, c, p Curly braces (brackets) { } Elements Set Notation Roster Method 2
- 3. { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9, pw, sw, c, p } Universal Set - U The set consisting of all of the elements to be considered. 3
- 4. A = { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9, pw, sw, c, p } B = { d, 3w, 5w, 7w } B is a subset of A B A A B One set (B) is a subset of another (A) if EVERY element of set B is contained in set A and is written B A 4
- 5. A = { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9} B = { 4, 5, 6, 7, 8, 9, pw, sw, c, p } {d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9 pw, sw, c, p }A B = Union of Sets Union: The set of elements that belong to either set A or set B or to both and is written A B 5
- 6. A = { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9} B = { 4, 5, 6, 7, 8, 9, pw, sw, c, p } { 4, 5, 6, 7, 8, 9 }A B = Intersection of Sets Intersection : The set of elements that belong to BOTH set A AND set B is denoted A B. 6
- 7. Null Set (empty Set) { } Not zero!!!! 7
- 8. Venn Diagrams Universal Set B A A B subset 8
- 11. Roster Method & Set-Builder Notation A = { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9, pw, sw, c, p } roster method A = { x | x is a club in the golf bag } set-builder notation Read “such that” 11
- 12. Roster Method & Set-Builder Notation A = { 6, 7, 8, 9, … } roster method A = { x | x is an integer > 5 } set-builder notation 12
- 13. Natural Numbers Natural or counting numbers. {1, 2, 3, …} Closed under addition. Example: 5 + 2 = 7 which is a natural number! 13
- 14. Natural Numbers Natural or counting numbers. {1, 2, 3, …} Closed under subtraction? Example: 2 – 5 = ? a natural number? 14
- 15. Integers {0, ±1, ± 2, ± 3, …} {0, ±1, ± 2, ± 3, …}{1, 2, 3,…} 15
- 16. Integers {0, ±1, ± 2, ± 3, …} Closed under addition & subtraction!Closed under multiplication?Closed under division? , , 16
- 17. Rational Numbers = { |m, n , n≠0} 17
- 18. Rational Numbers Closed under +, -, x, ÷ Denominator NEVER zero!! 18
- 19. Decimal expansions: Terminating ⅖ = .4 Or Non-terminating repeating ⅙ = .161616… Rational Numbers Expressed as Decimals 19
- 20. 3.1415926535897 932384626433832 795028841971693 993751058209749 445923078164062 862089986280348 2534211706798… Irrational Numbers An irrational number is a number that cannot be expressed as a fraction. Decimal expansions: Do NOT terminate. AND Do NOT repeat. Pi = π 20
- 21. is irrational ! 1.41421356237309504880168872420969807 8569671875376948073176679737990732478 4621070388503875343276415727350138… ≈ 1.41 21
- 22. Universe of Real Numbers Real Numbers 22
- 23. Real Number Line 1 2 3-1 0 π =U - 23
- 24. Real Number Line - Points Real Numbers1 to 1 -Is complete – (no holes) -Is ordered 24