1. Several sets are defined including sets of integers, rational numbers, and real numbers satisfying certain properties or conditions.
2. The concepts of elements, subsets, universal sets, and relative universes are explained. Finite and infinite sets are also introduced.
3. Examples are provided to demonstrate set operations and comparisons including union, intersection, subset, equality and inequality of sets. Properties of empty and singleton sets are also illustrated.
1. This document contains the text and work for exercises from Chapter 12 on three-dimensional space and vectors. It includes problems finding points and lines in 3D space, calculating distances, properties of spheres, triangles, and equations of spheres and ellipsoids.
2. Several questions involve finding distances between points in 3D space using the Pythagorean theorem, identifying right triangles, and calculating areas of triangles. Properties calculated include radii, diameters, centers and equations of spheres and ellipsoids.
3. Methods demonstrated include identifying perpendicular and parallel lines; locating points and lines in planes; using vectors; and applying equations, distances, areas and Pythagorean theorem to analyze three-dimensional geometric shapes and problems.
The document contains definitions of relations and functions through sets of ordered pairs. It includes relations defined by properties or equations, and functions defined as sets of inputs and outputs. It poses problems involving evaluating functions at given inputs, finding compositions of functions, and performing arithmetic operations on functions.
This document provides 85 essential revision questions for GCSE Mathematics at grades C to D. It covers topics such as:
- Expanding algebraic expressions
- Factorizing algebraic expressions
- Finding the nth term of patterns
- Calculating mean, median and mode from data
- Standard form and significant figures
- Prime factor decomposition
- 3D shapes from different views
- Solving equations
- Ratio and proportion word problems
The questions are arranged in order of difficulty from grades C to D. Answers are provided for self-checking. The purpose is to help students effectively revise key mathematical content for their GCSE exams.
This document contains a series of math problems and exercises related to place value, ordering numbers, rounding, reading scales, operations with decimals and fractions, percentages, and more. The problems are arranged into 13 clips or sections covering different math topics. Some of the key areas covered include:
- Writing numbers in figures and words
- Ordering and comparing decimals, fractions, percentages
- Rounding numbers to various places
- Reading information from scales and diagrams
- Performing calculations with positive and negative numbers
- Converting between fractions, decimals, and percentages
- Working with square and cube numbers
- Solving word problems involving money, time, and rates
The document provides a comprehensive set of math skills
1) The first integral evaluates to 4πi using the Cauchy integral formula applied to circles around z=1 and z=2.
2) The second integral evaluates the 4th derivative of e2z at z=-1 using a formula relating derivatives and contour integrals, giving a value of 24.
3) Both integrals are evaluated quickly using results from complex analysis without direct computation.
This document contains solutions to exercises from a chapter on partial derivatives. It includes:
1) Solutions to 14 sets of partial derivative exercises involving functions of two or more variables.
2) Discussion of limits of functions as the variables approach certain points, including cases where the limit does not exist.
3) Graphical representations of functions of two variables and their level curves.
The document provides detailed worked solutions to multiple partial derivative practice problems across several pages.
1. This document is a marking scheme for a trial examination paper for the Malaysian Certificate of Education (SPM) mathematics additional paper in 2009.
2. The marking scheme provides the full and sub marks awarded for each question on the paper. For multiple part questions, it outlines the marks given for each part. It also includes the expected solutions and working for many of the questions.
3. The paper consisted of 3 sections - Section A with 7 questions, Section B with 10 questions, and Section C with 5 questions. Most questions had multiple parts testing different math skills like algebra, geometry, calculus, and problem solving.
1. The document provides 13 mathematics questions covering topics such as: simplifying expressions, solving quadratic equations, working with cylinders and areas, and proving identities. Students must show their working and attempt all questions.
2. The questions involve skills like factorizing, using the quadratic formula, eliminating variables, finding minimum points on graphs, and proving statements about odd numbers. Working is required for full marks.
3. Students must complete the questions for homework and bring their work to the next mathematics lesson.
1. This document contains the text and work for exercises from Chapter 12 on three-dimensional space and vectors. It includes problems finding points and lines in 3D space, calculating distances, properties of spheres, triangles, and equations of spheres and ellipsoids.
2. Several questions involve finding distances between points in 3D space using the Pythagorean theorem, identifying right triangles, and calculating areas of triangles. Properties calculated include radii, diameters, centers and equations of spheres and ellipsoids.
3. Methods demonstrated include identifying perpendicular and parallel lines; locating points and lines in planes; using vectors; and applying equations, distances, areas and Pythagorean theorem to analyze three-dimensional geometric shapes and problems.
The document contains definitions of relations and functions through sets of ordered pairs. It includes relations defined by properties or equations, and functions defined as sets of inputs and outputs. It poses problems involving evaluating functions at given inputs, finding compositions of functions, and performing arithmetic operations on functions.
This document provides 85 essential revision questions for GCSE Mathematics at grades C to D. It covers topics such as:
- Expanding algebraic expressions
- Factorizing algebraic expressions
- Finding the nth term of patterns
- Calculating mean, median and mode from data
- Standard form and significant figures
- Prime factor decomposition
- 3D shapes from different views
- Solving equations
- Ratio and proportion word problems
The questions are arranged in order of difficulty from grades C to D. Answers are provided for self-checking. The purpose is to help students effectively revise key mathematical content for their GCSE exams.
This document contains a series of math problems and exercises related to place value, ordering numbers, rounding, reading scales, operations with decimals and fractions, percentages, and more. The problems are arranged into 13 clips or sections covering different math topics. Some of the key areas covered include:
- Writing numbers in figures and words
- Ordering and comparing decimals, fractions, percentages
- Rounding numbers to various places
- Reading information from scales and diagrams
- Performing calculations with positive and negative numbers
- Converting between fractions, decimals, and percentages
- Working with square and cube numbers
- Solving word problems involving money, time, and rates
The document provides a comprehensive set of math skills
1) The first integral evaluates to 4πi using the Cauchy integral formula applied to circles around z=1 and z=2.
2) The second integral evaluates the 4th derivative of e2z at z=-1 using a formula relating derivatives and contour integrals, giving a value of 24.
3) Both integrals are evaluated quickly using results from complex analysis without direct computation.
This document contains solutions to exercises from a chapter on partial derivatives. It includes:
1) Solutions to 14 sets of partial derivative exercises involving functions of two or more variables.
2) Discussion of limits of functions as the variables approach certain points, including cases where the limit does not exist.
3) Graphical representations of functions of two variables and their level curves.
The document provides detailed worked solutions to multiple partial derivative practice problems across several pages.
1. This document is a marking scheme for a trial examination paper for the Malaysian Certificate of Education (SPM) mathematics additional paper in 2009.
2. The marking scheme provides the full and sub marks awarded for each question on the paper. For multiple part questions, it outlines the marks given for each part. It also includes the expected solutions and working for many of the questions.
3. The paper consisted of 3 sections - Section A with 7 questions, Section B with 10 questions, and Section C with 5 questions. Most questions had multiple parts testing different math skills like algebra, geometry, calculus, and problem solving.
1. The document provides 13 mathematics questions covering topics such as: simplifying expressions, solving quadratic equations, working with cylinders and areas, and proving identities. Students must show their working and attempt all questions.
2. The questions involve skills like factorizing, using the quadratic formula, eliminating variables, finding minimum points on graphs, and proving statements about odd numbers. Working is required for full marks.
3. Students must complete the questions for homework and bring their work to the next mathematics lesson.
The document is the cover page of a mathematics question paper containing instructions and details about the exam. It states that the exam is for 2 1/2 hours with a maximum of 100 marks. The paper contains four sections. It instructs students to check for fairness of printing and inform the supervisor if any issues are found.
M A T H E M A T I C S I I I J N T U M O D E L P A P E R{Wwwguest3f9c6b
This document contains a mathematics examination for an engineering course. It consists of 8 questions testing various topics in complex analysis, including:
1) Evaluating integrals using techniques like Cauchy's integral formula and the residue theorem.
2) Finding Taylor and Laurent series expansions.
3) Identifying poles and residues of complex functions.
4) Applying complex transformations to map regions in the z-plane to the w-plane.
The questions cover topics such as contour integration, power series, analytic functions, and bilinear transformations. Students are instructed to answer any 5 of the 8 questions, each worth equal marks, in their 3-hour examination.
This document contains information about a mathematics exam for Form 5 students at Tunku Besar Burhanuddin Secondary School in Negeri Sembilan, Malaysia. The exam will take place in May 2010 and consists of 40 multiple choice questions to be completed in 1 hour and 15 minutes. The document provides formulas and concepts that may be useful for answering the questions. It also contains instructions for students on how to fill in the answer sheet.
The document contains examples of algebraic operations involving multiplication and division of polynomials and terms. Some key examples include:
- Simplifying expressions like (x + 2y + (x - y)) by distributing terms
- Combining like terms, such as 4m - 2n + 3 - (-m + n) + (2m - n) = 4m - 2n - 3 + (-m + n) - (2m - n)
- Multiplying polynomials following standard order of operations, such as a2b3 • 3a2x = 3a4b3x
- Dividing polynomials results in subtraction of exponents, like -xmync • -xmyncx = -
The document discusses spherical Bessel functions of fractional order. It defines the spherical Bessel functions of the first kind jn(z), the second kind yn(z), and the third kind hn(z). It provides representations of these functions by elementary functions, ascending series, Poisson's integral formula, and Gegenbauer's generalization. It also discusses properties such as differentiation formulae, analytic continuation, and generating functions. Tables are provided with values of the modified spherical Bessel functions for different orders and arguments.
This document contains 26 multiple choice questions about quadratic equations. The questions cover a range of topics including finding the roots of quadratic equations, determining the nature of the roots based on coefficients, and other properties of quadratic equations. Sample answers are provided but no full solutions are shown.
The document appears to be an exam paper for an Introduction to Information Technology course. It contains:
1) A 30 mark Section A with 15 short answer questions testing core IT concepts like operating systems, Internet, printers, memory etc.
2) A 45 mark Section B with 9 longer answer questions testing topics like computer organization, printers, DOS commands, LAN/WAN differences, applications of computers etc.
3) The paper is for a 3 hour exam with a maximum of 75 marks and instructs students to answer Section A compulsory and any 9 questions from Section B.
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems. We started to develop ways to enhance students IQ. We started to leave an indelible mark on the students who have undergone APEX training. That is why APEX INSTITUTE is very well known of its quality of education
1) The document discusses representing complex numbers geometrically using the Argand diagram. Complex numbers a + ib can be represented as a point (a,b) on the Argand plane, with the real part a on the x-axis and imaginary part b on the y-axis.
2) Examples are given of representing different complex numbers as points on the Argand plane, such as 2 + 3i as point (2,3). It is shown that a + bi is not the same as -a - bi, a - bi, or -z.
3) The modulus (absolute value) of a complex number a + ib is defined as the distance from the point (a,b) representing
210502 Mathematical Foundation Of Computer Scienceguestd436758
This document contains an exam for a Mathematical Foundations of Computer Science course, with 8 questions covering topics like propositional logic, functions, graphs, and algorithms. It provides the full exam paper, with diagrams and multiple parts to each question. The exam tests students on their understanding of key mathematical concepts that form the foundation for computer science.
This document describes a novel steganographic method for hiding data in JPEG images. It proposes improvements to the existing matrix encoding (F5) technique. Specifically, it introduces overlapped matrix encoding, modified matrix encoding using 2 or 3 coefficient flips instead of 1, and an insert-remove approach. Experimental results show this method achieves higher data hiding capacity while decreasing detectability compared to the original F5 technique according to steganalysis using 274 features.
This document contains:
1) An announcement about an assigned problem set due November 28th and office hours.
2) A summary of the regression theorem for finding local maxima, minima, and saddle points of functions with two variables.
3) An example of classifying critical points of a function.
4) A discussion of finding the line of best fit to a set of data points by minimizing the sum of squared errors between the data points and fitted line.
The document discusses finding the square of a binomial expression. It explains that to find the square of (a + b), the expression is (a + b)(a + b), not (a + b)2. Using FOIL or the distributive property, the square of (a + b) is a2 + 2ab + b2. Similarly, the square of (a - b) is a2 - 2ab + b2. The document provides examples of expanding squared binomial expressions and warns students not to make mistakes like (x + 6)2 = x2 + 36.
The document contains examples of functions of several variables and their domains and ranges. It provides equations for various functions and graphs their surfaces over different domains. Some key examples include functions defined by equations like x2 + y2 = 1, 2, 3 and functions where increasing one variable by a fixed amount increases the output by a fixed amount.
1. The decimal expansion of π is non-terminating and recurring.
2. If one diagonal of a trapezium divides the other in the ratio 1:3, then one of the parallel sides is three times the other.
3. The mode of the data with classes 50-60, 60-70, 70-80, 80-90, 90-100 and frequencies 9, 12, 20, 11, 10 respectively is 70-80.
This document contains exercises related to mathematics including:
1) Questions about relations being reflexive, symmetric, and transitive.
2) Matrix addition, multiplication, and inverse operations.
3) Questions involving functions, their compositions and inverses.
4) Questions testing knowledge of properties like commutativity and associativity.
The document provides problems and answers related to foundational mathematics concepts.
This document contains instructions and diagrams for 8 geometry problems involving calculating areas and perimeters of sectors and shaded regions of circles. The problems provide measurements for arc lengths and central angles of the sectors and ask students to use a given value of pi to calculate the requested values, showing their working and rounding answers to two decimal places. Marking schemes are provided for each multi-part problem.
1. The document contains 20 problems involving matrix algebra concepts such as determinants, inverses, eigenvalues, and system of linear equations. Sample solutions are provided for each problem.
2. Various matrix operations are introduced such as finding the determinant, inverse, and adjoint of matrices. Properties of these operations like (AB)-1 = B-1A-1 are explained.
3. Examples involve setting up and solving systems of linear equations represented by matrices, computing eigenvalues and eigenvectors, and exploring properties of matrix addition, multiplication, and transposition.
The document contains 23 math problems involving equations, inequalities, geometry concepts like angles and lengths of lines, limits, and other algebraic expressions. The problems cover a wide range of math topics including functions, polynomials, systems of equations, trigonometry, and calculus.
The document discusses sets and their properties. It defines sets like A={m,p,r,w}, B={2,1,0,1,2}, and C={2,3,5}. It calculates properties of sets like the number of elements (n(A)=4, n(B)=5, n(C)=3). It also discusses power sets and their sizes compared to the original sets, like P(A) having 16 elements while A only has 4. Subsets, proper subsets, and empty/null sets are also covered.
The document is the cover page of a mathematics question paper containing instructions and details about the exam. It states that the exam is for 2 1/2 hours with a maximum of 100 marks. The paper contains four sections. It instructs students to check for fairness of printing and inform the supervisor if any issues are found.
M A T H E M A T I C S I I I J N T U M O D E L P A P E R{Wwwguest3f9c6b
This document contains a mathematics examination for an engineering course. It consists of 8 questions testing various topics in complex analysis, including:
1) Evaluating integrals using techniques like Cauchy's integral formula and the residue theorem.
2) Finding Taylor and Laurent series expansions.
3) Identifying poles and residues of complex functions.
4) Applying complex transformations to map regions in the z-plane to the w-plane.
The questions cover topics such as contour integration, power series, analytic functions, and bilinear transformations. Students are instructed to answer any 5 of the 8 questions, each worth equal marks, in their 3-hour examination.
This document contains information about a mathematics exam for Form 5 students at Tunku Besar Burhanuddin Secondary School in Negeri Sembilan, Malaysia. The exam will take place in May 2010 and consists of 40 multiple choice questions to be completed in 1 hour and 15 minutes. The document provides formulas and concepts that may be useful for answering the questions. It also contains instructions for students on how to fill in the answer sheet.
The document contains examples of algebraic operations involving multiplication and division of polynomials and terms. Some key examples include:
- Simplifying expressions like (x + 2y + (x - y)) by distributing terms
- Combining like terms, such as 4m - 2n + 3 - (-m + n) + (2m - n) = 4m - 2n - 3 + (-m + n) - (2m - n)
- Multiplying polynomials following standard order of operations, such as a2b3 • 3a2x = 3a4b3x
- Dividing polynomials results in subtraction of exponents, like -xmync • -xmyncx = -
The document discusses spherical Bessel functions of fractional order. It defines the spherical Bessel functions of the first kind jn(z), the second kind yn(z), and the third kind hn(z). It provides representations of these functions by elementary functions, ascending series, Poisson's integral formula, and Gegenbauer's generalization. It also discusses properties such as differentiation formulae, analytic continuation, and generating functions. Tables are provided with values of the modified spherical Bessel functions for different orders and arguments.
This document contains 26 multiple choice questions about quadratic equations. The questions cover a range of topics including finding the roots of quadratic equations, determining the nature of the roots based on coefficients, and other properties of quadratic equations. Sample answers are provided but no full solutions are shown.
The document appears to be an exam paper for an Introduction to Information Technology course. It contains:
1) A 30 mark Section A with 15 short answer questions testing core IT concepts like operating systems, Internet, printers, memory etc.
2) A 45 mark Section B with 9 longer answer questions testing topics like computer organization, printers, DOS commands, LAN/WAN differences, applications of computers etc.
3) The paper is for a 3 hour exam with a maximum of 75 marks and instructs students to answer Section A compulsory and any 9 questions from Section B.
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems. We started to develop ways to enhance students IQ. We started to leave an indelible mark on the students who have undergone APEX training. That is why APEX INSTITUTE is very well known of its quality of education
1) The document discusses representing complex numbers geometrically using the Argand diagram. Complex numbers a + ib can be represented as a point (a,b) on the Argand plane, with the real part a on the x-axis and imaginary part b on the y-axis.
2) Examples are given of representing different complex numbers as points on the Argand plane, such as 2 + 3i as point (2,3). It is shown that a + bi is not the same as -a - bi, a - bi, or -z.
3) The modulus (absolute value) of a complex number a + ib is defined as the distance from the point (a,b) representing
210502 Mathematical Foundation Of Computer Scienceguestd436758
This document contains an exam for a Mathematical Foundations of Computer Science course, with 8 questions covering topics like propositional logic, functions, graphs, and algorithms. It provides the full exam paper, with diagrams and multiple parts to each question. The exam tests students on their understanding of key mathematical concepts that form the foundation for computer science.
This document describes a novel steganographic method for hiding data in JPEG images. It proposes improvements to the existing matrix encoding (F5) technique. Specifically, it introduces overlapped matrix encoding, modified matrix encoding using 2 or 3 coefficient flips instead of 1, and an insert-remove approach. Experimental results show this method achieves higher data hiding capacity while decreasing detectability compared to the original F5 technique according to steganalysis using 274 features.
This document contains:
1) An announcement about an assigned problem set due November 28th and office hours.
2) A summary of the regression theorem for finding local maxima, minima, and saddle points of functions with two variables.
3) An example of classifying critical points of a function.
4) A discussion of finding the line of best fit to a set of data points by minimizing the sum of squared errors between the data points and fitted line.
The document discusses finding the square of a binomial expression. It explains that to find the square of (a + b), the expression is (a + b)(a + b), not (a + b)2. Using FOIL or the distributive property, the square of (a + b) is a2 + 2ab + b2. Similarly, the square of (a - b) is a2 - 2ab + b2. The document provides examples of expanding squared binomial expressions and warns students not to make mistakes like (x + 6)2 = x2 + 36.
The document contains examples of functions of several variables and their domains and ranges. It provides equations for various functions and graphs their surfaces over different domains. Some key examples include functions defined by equations like x2 + y2 = 1, 2, 3 and functions where increasing one variable by a fixed amount increases the output by a fixed amount.
1. The decimal expansion of π is non-terminating and recurring.
2. If one diagonal of a trapezium divides the other in the ratio 1:3, then one of the parallel sides is three times the other.
3. The mode of the data with classes 50-60, 60-70, 70-80, 80-90, 90-100 and frequencies 9, 12, 20, 11, 10 respectively is 70-80.
This document contains exercises related to mathematics including:
1) Questions about relations being reflexive, symmetric, and transitive.
2) Matrix addition, multiplication, and inverse operations.
3) Questions involving functions, their compositions and inverses.
4) Questions testing knowledge of properties like commutativity and associativity.
The document provides problems and answers related to foundational mathematics concepts.
This document contains instructions and diagrams for 8 geometry problems involving calculating areas and perimeters of sectors and shaded regions of circles. The problems provide measurements for arc lengths and central angles of the sectors and ask students to use a given value of pi to calculate the requested values, showing their working and rounding answers to two decimal places. Marking schemes are provided for each multi-part problem.
1. The document contains 20 problems involving matrix algebra concepts such as determinants, inverses, eigenvalues, and system of linear equations. Sample solutions are provided for each problem.
2. Various matrix operations are introduced such as finding the determinant, inverse, and adjoint of matrices. Properties of these operations like (AB)-1 = B-1A-1 are explained.
3. Examples involve setting up and solving systems of linear equations represented by matrices, computing eigenvalues and eigenvectors, and exploring properties of matrix addition, multiplication, and transposition.
The document contains 23 math problems involving equations, inequalities, geometry concepts like angles and lengths of lines, limits, and other algebraic expressions. The problems cover a wide range of math topics including functions, polynomials, systems of equations, trigonometry, and calculus.
The document discusses sets and their properties. It defines sets like A={m,p,r,w}, B={2,1,0,1,2}, and C={2,3,5}. It calculates properties of sets like the number of elements (n(A)=4, n(B)=5, n(C)=3). It also discusses power sets and their sizes compared to the original sets, like P(A) having 16 elements while A only has 4. Subsets, proper subsets, and empty/null sets are also covered.
The document describes hierarchical clustering algorithms. It compares the single-link and complete-link algorithms. Single-link produces elongated clusters by connecting nearby points, while complete-link produces more compact clusters by only merging groups whose furthest points are close. Complete-link generally produces more useful hierarchies but is less versatile than single-link. Average linkage is also mentioned as an alternative that calculates distances between groups as the average of all point-point distances.
The document describes and compares different hierarchical clustering algorithms:
1) Single-link clustering connects clusters based on the closest pair of patterns, forming elongated clusters. Complete-link connects based on the furthest pair, forming more compact clusters.
2) Complete-link is more useful than single-link for most applications as it produces more interpretable hierarchies. However, single-link can extract certain cluster types that complete-link cannot, like concentric clusters.
3) Average group linkage connects clusters based on the average distance between all pairs of patterns in the two clusters. It provides a balance between single and complete link.
1. The document discusses geometric concepts such as lines, angles, and the Pythagorean theorem.
2. Equations and formulas are presented for calculating lengths of sides of right triangles based on the Pythagorean theorem.
3. Approximations of irrational numbers like the square root of 2 and pi are calculated through successive decimals.
1. The document discusses geometric concepts such as lines, angles, and the Pythagorean theorem.
2. Equations and formulas are presented for calculating lengths of sides of right triangles based on the Pythagorean theorem.
3. Approximations of irrational numbers like the square root of 2 and pi are calculated through successive decimals.
1. The determinant of a 3x3 matrix A is calculated to be 1.
2. The determinant of a 2x2 matrix B is calculated to be 1/3.
3. A system of 3 linear equations with 3 unknowns is solved using determinants to find the values of x, y, and z.
The document discusses sets and set operations including defining sets, elements, cardinal and ordinal numbers, equal and equivalent sets, subsets, Venn diagrams, and set operations like union, intersection, and complement. Examples are provided to illustrate concepts like finite and infinite sets, subsets, Venn diagrams representing multiple sets and operations, and using Venn diagrams to solve problems involving sets and their relationships.
LISTA DE EXERCÍCIOS - OPERAÇÕES COM NÚMEROS REAISwillianv
The document contains a series of math problems involving simplifying expressions, rationalizing denominators, factoring expressions, and simplifying fractions. It provides the problems and then lists the answers. Some of the problems include determining the values of expressions, simplifying radicals, factoring quadratic expressions, and calculating a value when given values for variables.
The document contains code snippets showing the use of variables, operators, control flow statements like if-else and switch-case, and input/output statements like printf and scanf in the C programming language. It defines variables, performs basic math operations on variables, and prints the results to the screen. Control structures like if/else and switch/case are used to check conditions and print corresponding outputs. Input is taken using scanf and output displayed with printf.
B.Sc (Pass) Nautical & Engineering Model Question 2 Mathematics Second Paper
(Differential Calculus, Integral Calculus, Two-dimensional & Three- dimensional Geometry)
The document provides examples and explanations of functions and relations. It uses graphs to demonstrate the vertical line test, which determines if a relation is a function. Several examples of relations are given and tested to see if they satisfy the vertical line test and are therefore functions. Practice problems are included for the reader to apply the concepts of domain, range, and determining if a relation represents a function.
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,aijcsa
A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of
“opposite edge strips” ops. Theta Θ(G,x) and Pi Π(G,x) polynomials can also be calculated by ops
counting. In this paper we compute these counting polynomials for a family of Benzenoid graphs that called
Hexagonal trapezoid system Tb,a.
This document contains a sample lesson plan on general mathematics for grade 11 that includes content on ordered pairs, functions, and determining whether a relation represents a function. The lesson includes examples of sets of ordered pairs and asks students to identify which are functions. It also contains graphs and asks students to determine if the relation is a function using the vertical line test. Key points covered are the definition of a function as a set of ordered pairs where no two pairs have the same first element and how to identify functions from graphs or sets of ordered pairs.
1. The document discusses geometry concepts including angles, lines, and polygons. It provides examples to illustrate properties such as angle-angle-side congruence, triangle sum theorem, and exterior angle theorem. Diagrams are included.
2. Algebraic expressions and equations involving variables are presented. Examples demonstrate solving linear equations, factoring polynomials, and properties of exponents.
3. Multiple proofs are given using logic and reasoning about the relationships between angles and sides of figures. Diagrams clearly label angles and sides to support the proofs.
The document reviews trigonometry concepts including the unit circle and finding trigonometric functions of special angles. It provides examples of the unit circle with coordinates marked around it and homework problems involving finding the trigonometric functions of various angles in radians and degrees. The review is intended to help remember things learned in trigonometry class.
1. The document is a study guide for a precalculus test that covers topics involving vectors, complex numbers, and their representations and operations.
2. It lists 15 problems involving finding magnitudes and direction of vectors, vector components, dot and cross products, complex numbers in rectangular and polar form, and converting between polar and rectangular coordinates.
3. The problems cover basic vector and complex number calculations, representations, properties and conversions.
4. 1.1
(Set)
19 (Georg Cantor)
“ ”
1.
“ ” “ ”
“ ” (Element)
1.
2.
3.
4.
5.
6.
“ (Universal set)”
U
I+ (positive integer number)
I- (negative integer number)
I ( integer number)
N (natural number)
P (prime number)
Q (rational number)
R (Real number )
5. R+ ( positive Real number)
R- ( negative Real number)
2.
A, B, C,…
a, b, c,…
2
1.
1.1 { }
A
A = {a, e, i, o, u} , a A a A
B
B={ , , }
C
C={ , , , , , , }
1.2
( ) 3
(…)
C 100
C = {1, 2, 3,…, 100}
B 53
B = {2, 4, 6,…, 52}
D
D={ , , ,…, }
1.3 ( )
3
A
A = {1,2,3,…}
E -1
E = {-1, -2, -3,…}
6. 3 ...
{3, 7, 5, 1, …}
1
2.
“ ”( , , such that)
1. A
A = {x x }
2. B
B={y y }
3. C
C = {z z }
4. E = {1,2,…,10}
E={x x 11}
5. F
F={x x }
1.
{1, 1, 2, 2, 2, 3} {1, 2, 3}
2.
{1, 2, 3} {3, 2, 1}
3.
3.
A = {2, 4, 6} 2, 4 6 A
7. 2 A 2 A
4 A 4 A
6 A 6 A
1 A 1 A
3 A 3 A
5 A 5 A
3.1
1. A = {3, 4, 5}
3 A, 4 A 5 A
7 A
2. B = {4, {5}}
4 B, {5} B
5 B
3. C = {{2, 3}, {4}}
{2, 3} C, {4} C,
2 C, 3 C 4 C
3.2 A = { 1, {2}, 3, {4, 5}}
1 A, {2} A, 3 A, {4,5} A
2 A, 4 A, 5 A
3.3 A = { a, b, {c}}
1. a A
2. b A
3. c A
4. {c} A
A 3 a, b {c}
1, 2 3, 4
8. 3.4 A = { 1, {3, 5}, 7 }
1. 1 { 1, {3, 5}, 7 }
2. 3 { 1, {3, 5}, 7 }
3. {3, 5} { 1, {3, 5}, 7 }
A 3 1, {3, 5} 7
1 3 2
n(A) A
3.5
1. A = {1, 2, 3}
A 3 n(A) = 3
2. B = {a, b, c, d, e}
B 5
3. C = {1, {2, 3}, {4}, 5, 6, 7}
C 6
(Empty set or null set)
{}
3.6
1. A =
A=
2. B = { x x x2 = -9 }
B= -9
3. C = { x x x x}
C= x x
9. 3.7 A 1 2
1 2 A=
B = {x x 5 - x = 6}
B=
C = {x x -2 -3}
-2 -3 C=
3.8
A={x x x2 = -1 }
B={x x }
C={x x –5 –7 }
A=
B=
C 1 -6
10. 1
1.
1)
2)
3) 3 10
4) 14
5) A = {x x = 2k + 1, k = 0, 1, 2, …}
6) B = {x x x2 – 3x + 2 = 0}
7) C = {y y y2 – 16 = 0}
8) D = {x x }
9) E = {x x }
10) G = {x x 3 10}
2.
1) A = { , , , }
2) C = { , , , }
3) B = {2, 4, 6, 8}
4) D = {…, -2, -1, 0, 1, 2,…}
5) E = {2, 4, 6, …}
6) F = {3, 6, 9, 12,…}
7) G = {5, 10, 15, 20,…, 100}
8) H = {-3, -2, -1, 0, 1, 2, 3}
9) I = {1, 4, 9, 16, 25, 36, …}
10) J = {1, 1 , 1 , 1 ,... }
2 3 4
3.
1) a {{a}}
2) 3 {{3}, 4}
3) {a} {{a}, b}
4) y {{x}, y}
5) x {{x, y}}
11. 4. B = {-2,{-1,0},1,{2,{3,4}},5}
1) 5 B
2) 2 B
3) {-1,0} B
4) {3,4} B
5) B
6) B B
5.
1) A = {-2, -1, 0, 1, 2, 3}
2) B = {{a, b}, 3}
3) C = {{1}, 2, {3, 4}, 5, 6}
4) D = {-1, {-2, -3, -4}, 0}
5) E = {a, b, c, {a, b}, {e, f, g}}
6) {x I | -2 x 5 }
7) {x I- | x + x = x 2 }
8) {x I- | x 12 }
9) {x | x = {1,2,3,4,…}}
10) {{}}
14. 1.2
4.
(Finite sets)
4.1
{x x x2 - x – 12 = 0}
{1, 2, 3, …,10}
4.2 M M
M={ , , , , , , }
M 7
M
4.3 B = { x I x2 0} B
B 0
B
4.4 A = { a, e, i, o, u }
B={x x x2 = 4 }
A B
A A 5
B B 2 2 -2
15. (Infinite set)
4.5
{1, 2, 3, …}
{x x }
{x x = 1 n N}
n 1
4.6 Y = { 1, 3, 5, …} Y
Y
Y
4.7 E = { x x 2 x 5} E
2 5
E
4.8
1. A = {1, 2, 5, 7}
4
2. B = {1, 2, 3, …}
3. {x x 1 }
12
4. {x x 0 < x < 1}
0 1
5.
A = {0, 1, 2, 3} B = {1, 0, 2, 3}
A B
16. A B A B
B A
A B A=B
A B A
B B A A
B A B
5.1
1) A = {5, 6, 7, 8} B = {5, 7, 6, 8}
A=B A B
2) A = {1, {1,2}} B = {1, 2}
A B A 1 {1, 2}
B 1 2
A B
5.2 A = {a, b, c}, B = { c, a, b, b}, C = {a, a, b, c, d}
A=B
A C B C
5.3
A = {5, 6, 7}
B = {y y (y – 5)(y – 6)(y – 7) = 0}
C = {y y 4 8}
D = {y y 8}
B, C D
B = {5, 6, 7}
C = {5, 6, 7}
D = {7, 6, 5, 4, 3, 2, 1, 0, -1, -2, …}
A = B, A = C, B = C, A D, B D C D
17. 5.4
A = { 1, 2, 4, 6, 6 }
B = { 1, 2, 4, 6 }
C={x x N 0 x 4}
D = { 1, 3, 5, 7 }
E = { 1, 2, 3 }
C
C = { 1, 2, 3 }
A=B A B
B A
C=E C E
E C
5.5 T = { 2, 4, 6}
S={x x 10 }
T S
S
S = { 2, 4, 6, 8 }
T S 8 S 8 T
{ 1, 1, 2, 3, 4, 4 } = { 1, 2, 3, 4 }
18. 6. (Relative Universe)
U
6.1
A 5 A
1, 2, 3, 4
U
B B
(2x - 1)(x + 4) = 0 -4
U
6.2 U
A = {x x2 = 4}
B = {x x3 = -1}
A = {-2, 2}
B = {-1}
U
A = {2}
B=
19. 2
1.
1) {x | x }
2) {x | x 3 }
3) {x I | x 2 0}
4) {x | x = 1 n }
n
5) {x x }
6) {x Q | 1 x 7 }
7) {x I- | x 2 x }
8) {x I- | x 12 }
9) {x | x = {1,2,3,4,…}}
2.
1) {x, x, x, y} {x, y}
2) {x, y} = {{x}, {y}}
3) {4} {{4}}
4) {3, 4} = {3, {3}, 4, {4}}
5) {0} = { }
3.
1) A = {4, 5, 6, 9} , B = { 9, 4, 6, 5}
2) C = , D = {0}
3) E = {x | x x }
20. 3
4
1
1.
2.
A B A B
A B
A A P (A)
3.
(Subset)
(power set )
4.
1.
2. 4 1.3
3.
4.
5. 1.3
6.
7.
22. 1.3
7. (Subset)
A B
A B A B
A B A B
A B A
B
A B A B
7.1 A= {0, 1, 2} , B = {0, 1, 2} , C = {3, 4, 5, 6} , D = {0, 1, 2, 3, 4, 5}
A B A B
B A B A
B D B D
A D A D
A C 1 A 1 C
B C 2 B 2 C
C D 6 C 6 D
A B A B B A
B A A=B
A B B A A=B
7.2 X = {a, b, c} Y = {c, a, b}
X Y X Y
Y X Y X
X=Y
23. A n A 2n
7.3 A = {1, 2, 3} A
A
1) {1} 5) {1, 3}
2) {2} 6) {2, 3}
3) {3} 7) {1, 2, 3} A
4) {1, 2} 8)
A 23 = 8
7.4 A = {1, 2}, B = {a, b, c} A B
A {1}, {2}, {1, 2}
A 22 = 4
B {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}
B 23 = 8
7.5 A = { 1, 2, 3, 4 }
1 {1}, {2}, {3}, {4}
2 {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}
3 {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}
4 {1, 2, 3, 4}
5
A 24 = 16
7.6 B = {1, {2}, {3}}
1 {1}, {{2}}, {{3}}
2 {1, {2}}, {1, {3}}, {{2}, {3}}
3 {1, {2}, {3}}
4
B 23 = 8
24. 7.7 A = {1, 2, 3}
B = {1, 2, 3, 4}
A B A B
B 4 B
A B A
7.8 A = {3, 4, 5}
B = {4, 3, 5}
A B A B
B A
B A A=B
7.9 A = {1, 3, 5, 7}
B = {3, 6, 9, 12, …}
C = {x x , x 3}
C A, B A B C
7.10
A={x x }
B={x x }
C={x x }
R={x x }
A B C R
7.11 A = {2, 4, 6}
B = {x x 1 x 7}
C = {2, 4, 6, 8}
A, B C B
B = {2, 4, 6}
25. A, B C
A B, A C
B A C A
1.
A A A
2.
A A
3.
A U A
8. (Power Sets)
A A
A P(A)
A n P(A) 2n
8.1 A = {5, 7, 9}
B = {a, b, {c}}
C = { , {1, 2 }},
D=
A, B, C D
P(A) = { {5}, {7}, {9}, {5, 7}, {5, 9}, {7, 9}, {5, 7, 9}, }
P(B) = { {a}, {b}, {{c}}, {a, b}, {a, {c}}, {b, {c}}, {a, b, {c}}, }
P(C) = { { }, {{1, 2}}, { , {1, 2}}, }
P(D) = { }
26. 8.2 A = { , 1, {2}} A
P(A) = { , { }, {1}, {{2}}, { , 1}, { , {2}}, {1, {2}}, { , 1, {2}}}
8.3 T = {a, b, c} T
P(T) = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
A B
1. A B P(A) P(B)
2. P(A B) = P(A) P(B)
3. P(A B) P(A) P(B)
A B
1. A
2. A A
3. A B B C A C
4. A B B A
5. A B A B
6. A = B A B B A
7. A P(A)
8. A P(A)
9. A B P(A) P(B)
10. P(A) P(B) P(A B)
11. P(A) P(B) = P(A B)
27. 3
1.
1) A=
2) B = {x}
3) C = {x, y}
4) D = {x, y, {a, b}}
5) E={ , , }
2.
1) {1}
2) {-1, 0, 1}
3) { , 0, {1}}
4) {0, {1}, {2}}
3.
1) {y y R y = y + 1}
2) {x x I 3 < x < 4}
3) {z z 5 }
4) { }
5) {0}
4. A
{}
A
A P(A)
A P(A)
P(A)
P(A)
28. 5. A = {a, {b, c}, b}
1) {0} A
2) {b} A
3) {{b, c}} A
4) {a, b} A
6. A = {-3,{2,a},b,{c,{1,4}},d}
1) -3 ….. A
2) {c,{1,4}} ….. A
3) {{2,a},d}….. A
4) {d,{c,{1,4}}} ….. A
5) {d, -3} ….. A
6) {-3,{2,a}} ….. A
7) A ….. A
8) ….. A
9) {d,b} ….. A
10) {b, {c,{1,4}} ….. A
11) {2, a} ….. A
12) {{c, {1, 4}}} ….. A
29. 4
- 4
1
-
1.
- (Venn-Euler Diagram)
2.
-
(U) U
3.
-
4.
1. 2 A B
1
U = { 1,2,3,4,…,10} A={2,4,5,6,8,9} B={1,3,4,5,7,10}
A
2. 4
1.4 -
3. 2
-
4. -
5. 4
31. 1.4
9. – (Venn – Euler Diagram)
–
U
U
A A B
A
B
U
U C U
1 2
A B
U
3
1 2 A, B C U
1 B A
2 A, B C
3 A B
(disjoint sets)
32. 9.1 A = {1, 2, 3, 4} B = {3, 4, 5, 6}
-
A B 3 4 -
A B
A
B
1 3 5
2 4 6
U
9.2 A = {1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8}
-
A B 2, 4 6 -
A B
A B
1 2
3 4 8
6
5
U
A A
B B
1 2 2
3 4 8 4
5 6 6
U U
A B A B
A A
B B
1
8 3
5
U U
A A-B
33. 4
1. U
1) A = {1, 2, 3,…,10}
B = {1, 3, 5, 7, 9}
2) A = {1, 2, 3,…,10}
B = {1, 3, 5, 7, 9}
C = {1, 3, 5}
2. U = {a, b, c, d, e, f, g, h, x, y}
A = {a, b, c, x}
B = {b, c, d, e}
C = {c, d, f, x}
3. U = {1,2,3,…,10}
A = {1, 3, 5, 7, 9}
B = {2, 4, 6, 8}
C = {2, 6}
D = {1, 3, 4}
4.
A B
B C
A C
34. 5
4
2
1.
2.
(Union) A B A
B A B
( intersection ) A B
A B A B
( complement ) A
U A A
(difference) A B A
B A-B
3.
4.
1. 4
1.5
2. 2
2
3.
4. 1
2-4
36. 1.5
10.
4
1. (union)
2. (intersection)
3. (complement)
4. (difference)
1. (union)
A B A
B
A B A B
A B = {x x A x B x }
A B –
A B
U
A B
37. 10.1 A = {1, 2, 3, 4} B = {3, 4, 5}
A B = {1, 2, 3, 4, 5}+
10.2 C = {a, b, c, d, e} D = {f, g, h, e, d}
C D = {a, b, c, d, e, f, g, h}
10.3 M = {1, 2, 3, 4,…} N = {1, 2, 3, 4}
M N = {1, 2, 3, 4,…}
10.4 A = {1, 2, 3, {4}} N = {1, {2, 3}, 4}
A B = {1, 2, 3, 4, {4}, {2, 3}}
2 2
1. A A=A
2. A B=B A
3. A (B C) = (A B) C
4. A =A : A U=U
5. A B A B=B
2. (intersection)
A B
A B
A B A B
A B = {x x A x B}
38. A B –
A
A B
U
U
A B
10.5 A = {0, 1, 2, 3} B = {0, 3, 5}
A B = {0, 3}
10.6 A = {0, 1, 2, 3}, B = {0, 1, 2, 3, 4} C = {0}
A B = {0, 1, 2, 3},
A C = {0}
B C = {0}
10.7 A = {0, 1, 2, 3} B = {7, 8, 9}
A B=
10.8 A = {2, 4, 6, 8} B=
A B =
10.9 A = {1, {1, 4}, , { }}
B = {1, { }, 5}
A B = {1, { }}
1. A A = A
2. A B = B A
3. A (B C) = (A B) C
4. A = :A U=A
5. A B A B=A
39. 5
1 1 1
1. U = 1, 1 , 1 , ,1,1, 1 , ,
2 3 4 5 6 7 8 9
A= 1 , 1 , 1, 1
5 6 7 8
B = 1, 1 , 1 , 1
2 5 7
C = 1, 1 , 1
4 3
1) A B B A
A B =……………………………………………………………………………
B A = …………………………………………………………………………….
A B ……………B A
2) A B B A
A B =……………………………………………………………………………
B A =……………………………………………………………………………
A B ……….. B A
3) B C =……………………………………………………………………………
A ( B C ) =……………………………………………………………………
(A B ) C =……………………………………………………………………
A ( B C )…………… (A B ) C
4) B C =…………………………………………………………………………….
A ( B C ) =……………………………………………………………………
A C =………………………………………………………………………….
(A B ) (A C) =……………………………………………………………
A ( B C )………. (A B ) ( A C )
5) A A =…………………………………………………………………………….
A =……………………………………………………………………………
A A…………. A
6) A A =……………………………………………………………………………..
A =……………………………………………………………………………
A A ………… A
7) A ( A A ) =……………………………………………………………………..
A ( A A ) =………………………………………………………………………
40. 2.
1) A,A U = A
…………………………………………………………………………………………………
………………………………………………………………………………………………
2) A,A U = A
…………………………………………………………………………………………………
…...……………………………………………………………………………………………
………...………………………………………………………………………………………
3) A,A = A
…………………………………………………………………………………………………
…………………………………………………………………………………………………
4) A,A =A
…………………………………………………………………………………………………
…………………………………………………………………………………………………
3. A B
1) A B
=…………………………………………………..……………………………………………
………………………………………………………………………………………………..
2) A B
=…………………………………………………………….…………………………………
……………………………………………………………………………………………….
41. 1.6
3. (complement)
U A U
A
A A
A = {x x U x A}
A –
A
U
A
A Ac , A , C(A)
10.10 U = {0, 1, 2, 3, 4, 5}, B = {0, 2, 4} C = {3, 4}
B = {1, 3, 5}
C = {0, 1, 2, 5}
10.11 U = {0, 1, 2, 3, 4}, A = {0, 2, 4} B = {3, 4}
A = {1, 3}
B = {0, 1, 2}
42. 10.12 U = {1, 2, 3, 4, 5, 6}, A = {3, 4, 6} , B = {5, 6}
C = {1, 2, 4, 5}
A = {1, 2, 5}
B = {1, 2, 3, 4}
C = {3, 6}
10.13 U = {1, 2, 3, …, 10}
A = {x x = 2n, n N 1 n 5}
A = {1, 3, 5, 7, 9}
4. (difference)
A B A B
A B A-B
A - B = {x x A x B}
A-B –
A B
U
A-B
A = U-A
10.14 A = {1, 2, 3, 4} B = {2, 4, 5, 6}
A - B = {1, 3}
43. 10.15 A = {a, b, c} B = {x, y}
A - B = {a, b, c}
10.16 A = {3, 7} B = {3, 7, 8}
A-B =
10.17 U = {1, 2, 3, …, 10}
A = {1, 3 , 5, 7, 9}
B = {3, 6, 9}
C = {3, 4, 8}
1) A
2) A (B C)
3) A–(B C)
4) (A B)-(C B)
1) A = U- A
= {1, 2, 3, …, 10} - {1, 3 , 5, 7, 9}
= {2, 4, 6, 8, 10}
A = {2, 4, 6, 8, 10}
2) A (B C ) = {1, 3 , 5, 7, 9} ( {3, 6, 9} {3, 4, 8} )
= {1, 3 , 5, 7, 9} {3}
= {1, 3 , 5, 7, 9}
A ( B C ) = {1, 3 , 5, 7, 9}
3) A – ( B C ) = {1, 3 , 5, 7, 9} – ( {3, 6, 9} {3, 4, 8} )
= {1, 3 , 5, 7, 9} - {3}
= {1, 5, 7, 9}
A – ( B C ) = {1, 5, 7, 9}
4) ( A B ) - ( C B ) = ( {1, 3 , 5, 7, 9} {3, 6, 9} ) – ( {1, 2, 5, 6, 7, 9, 10}
{3, 6, 9} )
= {3, 9} – {1, 2, 3, 5, 6, 7, 9, 10}
=
(A B)-(C B) =
44. 1. (A ) = A
2. =U U =
3. (A B) = A B
(A B) = A B
4. A A =
A A =U
5. A – B = A B
6. A B A–B =
10.18 U = {0, 1, 2, {2}, {1, 2}}
A = {0, 1, 2}
B = {1, 2 {2}}
1. (A B )–A=B
2. A – (A B) = A - B
3. (A B ) (A B) = A B
4. P(A) P(B)
–
U
A B
1
0 {2}
2
{1,2}
1. (A B )–A = {{2}, {1, 2} {0, {1, 2}} – {0, 1, 2}
= {{1, 2}} - {0, 1, 2}
(A B ) – A = {{1, 2}}
B = {0, {1, 2}}
(A B )–A B
45. 2. A – (A B) = {0, 1, 2} - {1, 2}
A – (A B) = {0}
A – B = {0, 1, 2} – {1, 2, {2}}
A – B = {0}
A – (A B) = A – B
3. (A B) (A B) = A B
(A B ) = {0, 1, 2} {0, {1, 2}}
= {0}
(A B) = {{2}, {1, 2}} {1, 2, {2}}
= {{2}}
(A B ) (A B) = {0, {2}}
A B = {0, 1, 2} {1, 2, {2}}
= {0, 1, 2, {2}}
(A B ) (A B) A B
4. B = {1, 2, {2}} A = {0, 1, 2}
B A
P(B) P(A)
A, B C
1. A (B C) = (A B) C
2. A (B C) = (A B) C
3. A B = B A
4. A B = B A
5. A (B C) = (A B) (A C)
6. A (B C) = (A B) (A C)
7. (A B) = A B
8. (A B) = A B
9. A A = U
10. A A =
11. (A ) = A
46. 6
1. U= -10,-9,-8,…0
A= -9 ,-7 ,- 6,-1, 0
B= -8, -7 ,-4 , -2, 0
C= -8 ,-6 ,-5 , -3 , 0
D= -10 ,-9 ,-8 ,-5
1) A ( B C )
2) (A B ) C
3) ( B C )
4) ( A B ) ( A C )
5) A - ( B- C )
6) (A - B ) –C
7) A -( B C )
8) (A - B ) (A - C )
9) A ( B- C )
10) ( A B ) - ( A C )
11) ( A B ) ( C - D )
12) ( A B ) - ( C D )
2. U= x x
A= y y
B= z z
C= w w
D= t t
A,B,C D
3. A = 1 , 2, 3
1) U = 0, 1, 2, 3, 4, 5 A
2) U = 1 , 2, 3 A
47. 4. A U
1) A A
………………………………………………………………………………………………………
2) A U
………………………………………………………………………………………………………
3) A
………………………………………………………………………………………………………
4) A U
………………………………………………………………………………………………………
5) U
………………………………………………………………………………………………………
6)
………………………………………………………………………………………………………
7) A - A
………………………………………………………………………………………………………
8) A A
………………………………………………………………………………………………………
9) U A’
………………………………………………………………………………………………………
5.
1) A B 2) A B
A B U
U A B U
3) A B 4) ( A B ) C
A
A B
B U A
A U
C
C
B
B
B
48. 5) (A B ) (B C)
U
U
A
C
C B
6) ( A B ) C
A A U
B C
C
7) ( A C ) ( B C )
A
A U
U
A
B C
49. 6
4
1
1.
2.
1 n A n U nA
2 n A B n A n A B
3 n A B n A n B n A B
4 nA B C nA nB nC nA B nA C nB C nA B C
3.
4.
1. 4 1.7
2.
3. 1
4.
( )
5.
6. 7
51. 1.7
A A
n(A)
A B A B= A B
A B A B
n (A )= m n (B) =k
n( A B ) = m + k = n( A )+ n( B)
m B
A k
U
1 A B
A B= n(A B) = n(A) + n(B)
1 A = 0 , 1 , 2, 3 B= 4,5,6
A B = A B = 0,1,2,3,4,5,6
n (A B) = 7
n (A) + n (B) = 4 + 3 = 7 n(A B) = n (A) + n (B)
2
2 A 1 , A 2 , …Am
n (A 1 A 2 … Am) = n (A 1 ) + n(A 2 ) + …+n (Am)
52. 2 A = 0,1,2 , B= 3,4 ,5,6,7 C= 8,9
A B = A C =B C =
A,B,C
A B C = 0,1,2,3,4,5,6,7,8,9
n (A B C) = 10
n(A) + n (B )+ n (C) = 3 + 5 + 2 = 10
n (A B C) = n(A) + n (B )+ n (C)
n(A B) A B
A B
n( A ) = m , n( B ) = k n (A B ) = s
1 A B= n (A B ) = s = 0
1
n(A B) = n( A ) + n( B ) = m + k = m + k - s
n(A B) = n( A ) + n( B ) – n (A B)
2 A B n(A B) = s 0
U
A B
n (A - B) = m – s
n (B - A) = k – s
n(A B) = s
A–B ,B–A A B
A B = (A-B) (B-A) (A B)
n (A B) = n((A-B) (B-A) (A B))
= n(A-B) + n(B-A) + n(A B)
= (m - s) + (k – s )+ s
= m+k–s
n (A B) = n(A) + n(B) – n (A B)
53. 3 A B
n (A B) = n(A) + n(B) – n (A B)
3 A= 0,1,2,3,4 , B= 3,4,5,6
A B = 0,1,2,3,4,5,6
n (A B) = 7 ………….( 1 )
n(A) = 5 , n(B) = 4 n (A B) = 2
n(A) + n(B) – n (A B) = 5 + 4 – 2 = 7 ………….( 2 )
(1) (2)
n (A B) = n(A) + n(B) – n (A B)
A, B C
n(A B C) 3
n(A B C) = n(A (B C))
= n(A) + n(B C) - n(A (B C))
= n(A) + n(B) + n(C) - n(B C ) - n((A B) (A C ))
= n(A) + n(B) + n(C) - n(B C ) - n(A B) + n(A C)
- n((A B) (A C ))
= n(A) + n(B) + n(C) - n(B C ) - n(A B) - n(A C)
+ n(A B C)
4 A,B C
n(A B C) = n(A) + n(B) + n(C) - n(A B) - (A C )
- n(B C) + n(A B C)
4
A,B,C D
54. n(A B C D) = n(A) + n(B) + n(C) + n(D) - n(A B) - n(A C )- n(A D )
- n(B C) - n(B D) - n(C D) + n(A B C)
+ n(A B D) + n(A C D) + n(B C D)
- n(A B C D)
A B 1 A–B
B–A
A A-B A B-A B
U
B
U
A=(A–B) (A B) (A–B) (A B) =
1
n (A) = n((A – B ) ( A B ) )
= n(A – B ) +n( A B )
n(A – B ) = n(A) - n( A B ) ………………( 1 )
n(B – A ) = n(B) - n( A B ) ………………( 2 )
(1) (2)
5 A B
n ( A – B ) = n (A) – n (A B)
n ( B – A ) = n (B) – n (A B)
A U U n(A)
n( U ) n(A ) 1
U
A
A
U=A A A A = 1
n( U ) = n(A A ) = n (A)+(A )
n (A ) = n(U ) - n (A)
55. 6 A U U
n(A ) = n( U)-n(A)
4 A B U n( U) = 100,
n( A) = 60 , n(B) = 75 n(A B) = 45
( 1 ) n(A B) (2) n(A ' B ' )
( 3 ) n(A ' B ' ) (4) n(A - B )
( 5 ) n(B A ' ) (6) n (A ' )
( 7 ) n(B ' ) (8) n(A B ' )
( 9 ) n( B A ' ) (10) n( A ' B ' )
( 1 ) n(A B) = n ( A ) + n ( B ) – n (A B)
= 60 + 75 – 45
= 90
( 2 ) n(A ' B ' ) = n ((A B) ' )
= n (U) - n (A B)
= 100 – 90
= 10
( 3 ) n(A ' B ' ) = n ((A B) ' )
= n (U) - n (A B)
= 100 – 45
= 55
( 4 ) n(A - B ) = n (A) - n(A B)
= 60 – 45
= 15
56. ( 5 ) n(B A ' ) = n(B - A )
= n (B) - n(A B)
= 75 – 45
= 30
( 6 ) n (A ' ) = n (U) - n (A)
= 100 – 60
= 40
( 7 ) n (B ' ) = n (U) - n (B)
= 100 – 75
= 25
( 8 ) n(A B ' ) = n(A) + n(B) - n(A B)
= 60 +25 – 15
= 70
( 9 ) n ( B A ' ) = n(B) +n(A) - n(B A)
= 75 + 40 – 30
= 85
(10) n ( A ' B ' ) = n ( B – A )
= 30
5 A,B C U n( U) = 100,
n(A) = 29 , n(B) = 23 , n(C) = 18 , n(A B) = 15 ,n (A C) = 10 , n(B C) = 9
n(A B C) = 6
( 1 ) n (A B ) ( 2 ) n (B ' C ' )
( 3 ) n ( A C') ( 4 ) n (A B ' )
( 5 ) n(A B C) ( 6 ) n(A ' B ' C ' )
( 7 ) n(A B C ' ) ( 8 ) n(A B ' C ' )
( 1 ) n (A B ) = n (A) + n(B) – n (A B)
= 29 + 23 –15
= 37
57. ( 2 ) n (B ' C ' ) = n ((B C) ' )
= n (U) – n(B C)
= n (U) - n (B) + n(C) - n(B C)
= 100 – 23 + 18 – 19
= 100 – 32 = 68
( 3 ) n ( A C ' ) = n(A) + n(C ' ) - n(A C ' )
= n(A) + n(U) - n(C) - n(A) - n(A C)
= 29 + (100 – 18 ) – ( 29 – 10 )
= 29 + 82 – 19 = 92
( 4 ) n (A B ' ) = n (A - B)
= n (A) – n (A B)
= 29 –15 = 14
( 5 ) n(A B C) = n (A) + n(B) + n(C) - n(A B)- n(A C)
- n(B C) + n(A B C)
= 29 + 23 + 18 – 15 – 10 - 9 + 6
= 42
( 6 ) n(A ' B' C ' ) = n ((A B C) ' )
= n(U) – n(A B C)
= 100 - 42 = 58
( 7 ) n(A B C ' ) = n ((A B) - C)
= n(A B) - n(A B C)
= 15 – 6 = 9
( 8 ) n(A B ' C ' ) = n ((A B ' ) - C)
= n (A B) - n(A B ' C)
= 14 – n(A C B ' )
= 14 - n(A C) - n(A C B)
= 14 - 10 –6
= 14 – 4 = 10
58. 7
1. A B U n (U) = 150 , n (A ) = 62 ,
n ( B ) = 55 n ( A B ) = 11
(1.1) n ( A B ) (1.2) n ( A B ) (1.3) n( A B ) (1.4) n ( A B )
(1.5) n( B A ) (1.6) n ( A ) (1.7) n ( B ) (1.8) n ( A B )
(1.9) n( B A ) (1.10) n ( A - B ) (1.11) n ( B - A )
2. A B U
n( u ) = 50, n ( A B ) = 6, n ( A B ) = 38 n(A) = n(B)
(2.1) n (A) (2.2) n (A ) (2.3) n (A – B) (2.4) n (B – A)
(2.5) n ( A B ) (2.6) n ( A B ) (2.7) n ( A B ) (2.8) n( B A )
(2.9) n ( A - B ) (2.10) n ( B - A )
3. A,B C U n ( U ) = 80 , n ( A ) = 35
n ( B ) = 28, n( C ) = 21 , n ( A B ) = 12 , n ( A C ) = 14 , n ( B C )=10
n (A B C ) = 4
(3.1) n ( A B ) (3.2) n( B C )
(3.3) n ( A C ) (3.4) n (A B C )
(3.5) n ( A B ) (3.6) n ( B C )
(3.7) n ( A C ) (3.8) n (A B C )
(3.9) n ( A B ) (3.10) n( B C )
(3.11) n ( A C ) (3.12) n (A B C )
(3.13) n ( A- B ) (3.14) n ( B - C )
(3.15) n ( C - A ) (3.16) n (A B C )
(3.17) n (A C B ) (3.18) n (B C A )
(3.19) n (A C B ) (3.20) n (B A C)
(3.21) n (C A B ) (3.22) n( ( A B )- C )
59. 4. A,B C U n (U ) = 100 ,
n ( A B C ) = 89 , n ( A B C ) = 5 , n ( A B )= 11, n ( A C)= 10, , n ( B C)= 9
n (A ) n (B ) n (C ) n (B )
(4.1) n (A ) (4.2) n (B )
(4.3) n (C ) (4.4) n ( A B C)
(4.5) n ( A B C ) (4.6) n ( A B C)
(4.7) n ( A C B ) (4.8) n ( B C A)
(4.9) n ( A C B ) (4.10) n ( B A C)
62. 1.8
1
55
38
22
U
A
B
A B
A B
n(A B)
n(A B)
n(A B) = 55, n(A) = 38
n(B) = 22
n (A B) = n(A) + n(B) - n(A B)
n(A B) = n(A) + n(B) - n (A B)
= 38 + 22 - 55 = 5
5
2 1,000
370
550 850
U
A
B
A B
63. A B
n(A) = 370
n(B) = 550
n(A B) = 850
x
A B
370 - x x 550 - x
U
n(A B) = x
n (A B) = n(A) + n(B) - n(A B)
850 = 370 + 550 - x
x = 370 + 550 - 850
= 70
70
3
A, B C 30%, 40% 50%
A B 10% A C 15% B C 20%
A, B C 3%
1) A, B C
2) A, B C
U A, B, C
A A
B B
C C
A B A B
A C A C
64. B C B C
A B C
A B C A, B C
(A B C) A, B C
100%
n(U) = 100
n(A) = 30
n(B) = 40
n(C) = 50
n(A B) = 10
n(A C) = 15
n(B C) = 20
n(A B C) = 3
C 18
12 17
3
A 8 13
7 B
U
1) A, B C
n(A B C) = n(A) + n(B) + n(C) – n(A B) - n(A C)
- n(B C) + n(A B C)
= 30 + 40 + 50 - 10 - 15 - 20 - 3
= 72
A, B C 72%
2) A, B C
A B C
100% A, B C
65. A B C n(A B C)
n(A B C) = 100% - 72% = 28%
A B C 28%
4 6 80
1
50 40 33
3 5 10 12
13
1)
2)
3)
U 6 80
A
B
C
B = 40
A = 50
20 15 12
5
10 8
10
C = 33
8
15
10
20
1)
50 + 12 + 8 = 70
2)
33 + 20 + 15 = 68
3)
40 + 10 + 10 = 60
68. 1
1. A = {x I 2x2 – 3x – 2} B = {x 3 x 4}
1. 2 A
1
2. A
2
3. 4 B
4. B
2. {0, 4, 6, 8, {4, 6}, 10}
1. 5
2. 6
3. 7
4. 32
3. A
1. X = { -1 , 0 , A }
2. A { -2 , -1 , 0 }
3. A -{ 5 , 6 , 7 }
4. A { -2 , -1 , 0 }
4.
1.{x R x + x = x}
2.{x R 5x = x2}
3.{x R 3 x x}
4.{x R x > x + 1}
5. A,B,C U
. A - ( B C) = ( A - B ) ( A - C )
. A - ( B C) = ( A - B ) ( A - C )
. ( A B) - C = ( A - C ) ( B - C )
1. , 3
2. ,
3.
4.
69. 6. A = { a , {a} , {b} ,{b,c} } (A - {b , c } {b})
1. { a , {a} , {b} }
2. { a , b , {a} }
3. { a , b ,{a} ,{b} }
4. { a , b , {a} , {b} , {b , c } }
7. B = { , 0 , 1 } P(B) B
1. P(B) 0 P(B)
2. P(B) 1 P(B)
3. { } P(B) {1} P(B)
4. { } P(B) {0} P(B)
8. A = { , 1, {1}, 2, 3, {1, 2}, { }, {{ }}}
. A . {1, 2} A
. {1, 2} A . { } A
. {{ }} A . {2, 3} A
1.
2. ,
3. ,
4.
9.
. A B B C A C
. A B A (A B)
. A B ,B C A C A B C
1.
2.
3.
4.
70. 10. -
1. (B C) A
A
A B
B
1. (B C) A
C
C U
U 2. (B C) A
3. (B C) A
11. U = {-3, -2, -1, 0, 1, 2, …, 10}
A = {1, 2, 3, 5, 7}
B = {3, 4, 5, 6, 7, 8, 9}
C = {-1, 0, 1, 2}
(A - B) (C A)
1. {1, 4, 3, 5, 7}
2. {1, 2, 3, 7, 9}
3. {1, 2, 3, 5, 7}
4. {1, 2, 3, 5, 7}
12. A = {0, 4, 8}, B = {1, 8, 5, 7, 9} C = {0, 2, 4, 6, 9}
A (B C)
1. 2. A
3. B 4. C
13. A = {1, 2, {1, 3}, 6, {2}} A
1. 5 2. 6
3. 12 4. 32
14. A = { , { }}, B = { , {{ }}}, C = { , { }, { , { }}}
. A B
. A C
. A C
1. ,
2.
3.
4. ,
71. 15.
. { , { }} - { } = { }
. { , { }} - {{ }} = { }
. { , { }} - = {{ }}
1. ,
2.
3.
4.
16. A, B, C
1. (A B) (A B) A B
2. A C=B C A=B
3. A B A B
4. B C A B A C
17.
. A B=A C B=C
. A B=A C B=C
. A –B = A – C B=C
. B–A=C–A B=C
1.
2.
3. ,
4.
18. U = {0, 1, 2, 3, 4, 5}, A B = { 0 , 1 , 2 , 3 }, A B = { 0 , 1 , 2 , 3 , 4 , 5 } ,
A C = { 0 , 1 }, B C = { 0 , 1 , 5 } A C={0,1,2,3,5} A' - (B C)
1. {0, 1, 3}
2.
3. { 0 , 1 , 5 }
4. {4}
72. 19.
. x C B C x B
. A -C =C–A
. [A (A B)] [B (A B)] A B
1.
2.
3.
4.
20. ผลการสำรวจของสาธารณสุขจังหวัด พบวาประชาชนเปนโรคตา 1,600 คน เปนโรคทางหู
20. 1,600
2,000 คน แตพบวาผูปวยมีรายชื่อซ้ำกันอยู 1,200 คน ผูปวยสองกลุมนี้มีจำนวนที่แทจริงกี่คน
2,000 1,200
1. 3,600
2. 3,200
3. 2,800
4. 2,400
21. ผลการสอบถามนั ก เรี ย น 1,000 คน พบว า 400 คนไม ช อบอาชี พ ครู 380 คนไม ช อบอาชี พ
21. 1,000 400 380
542 294 277
190
1. 143
2. 132
3. 88
4. 83
22. 748
1 3 348
338 312
204 210
196
1. 112
2. 106
3. 98
4. 84