Math

AlgebraElements Of Modern AlgebraIn each of the following parts, a relation R is defined on the set Z of all integers. Determine in each case whether or not R is reflexive, symmetric or transitive. Justify your answers. a. x R y if and only if x = 2 y . b. x R y if and only if x = − y . c. x R y if and only if y = x k for some k in Z . d. x R y if and only if x < y . e. x R y if and only if x ≥ y . f. x R y if and only if x = | y | . g. x R y if and only if | x | ≤ | y + 1 | . h. x R y if and only if x y ≥ 0 . i. x R y if and only if x y ≤ 0 . j. x R y if and only if | x − y | = 1 . k. x R y if and only if | x − y | < 1 .BuyFind*arrow_forward*

8th Edition

Gilbert + 2 others

Publisher: Cengage Learning,

ISBN: 9781285463230

Chapter 1.7, Problem 2E

Textbook Problem

In each of the following parts, a relation

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

Elements Of Modern Algebra

Show all chapter solutions

Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False
Label each of the following...Ch. 1.1 - True or False
Label each of the following...Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False
Label each of the following...Ch. 1.1 - True or False
Label each of the following...Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False Label each of the following...

Ch. 1.1 - For each set A, describe A by indicating a...Ch. 1.1 - 2. Decide whether or not each statement is true...Ch. 1.1 - Decide whether or not each statement is true. (a)...Ch. 1.1 - 4. Decide whether or not each of the following is...Ch. 1.1 - Evaluate each of the following sets, where U={...Ch. 1.1 - 6. Determine whether each of the following is...Ch. 1.1 - Write out the power set, (A), for each set A. a....Ch. 1.1 - 8. Describe two partitions of each of the...Ch. 1.1 - Write out all the different partitions of the...Ch. 1.1 - 10. Suppose the set has a .
a. How many elements...Ch. 1.1 - 11. State the most general conditions on the...Ch. 1.1 - 12. Let Z denote the set of all integers, and...Ch. 1.1 - 13. Let Z denote the set of all integers, and...Ch. 1.1 - In Exercises 1435, prove each statement. ABABCh. 1.1 - In Exercises 1435, prove each statement. (A)=ACh. 1.1 - In Exercises , prove each statement.
16. If and ,...Ch. 1.1 - In Exercises , prove each statement.
17. if and...Ch. 1.1 - In Exercises , prove each statement.
18.
Ch. 1.1 - In Exercises , prove each statement.
19.
Ch. 1.1 - In Exercises 1435, prove each statement. (AB)=ABCh. 1.1 - In Exercise 14-35, prove each statement.
21.
Ch. 1.1 - In Exercise 14-35, prove each statement. A(AB)=ABCh. 1.1 - In Exercises 14-35, prove each statement.
23.
Ch. 1.1 - In Exercise 14-35, prove each statement....Ch. 1.1 - In Exercise 14-35, prove each statement. If AB,...Ch. 1.1 - In Exercise 14-35, prove each statement.
26. If...Ch. 1.1 - In Exercise 14-35, prove each statement.
27.
Ch. 1.1 - In Exercise 14-35, prove each statement. A(BA)=Ch. 1.1 - In Exercises 14-35, prove each statement.
29.
Ch. 1.1 - In Exercises 14-35, prove each statement....Ch. 1.1 - In Exercises 1435, prove each statement....Ch. 1.1 - In Exercises 1435, prove each statement....Ch. 1.1 - In Exercises , prove each statement.
33.
Ch. 1.1 - In Exercises , prove each statement.
34. if and...Ch. 1.1 - In Exercises 1435, prove each statement. AB if and...Ch. 1.1 - Prove or disprove that AB=AC implies B=C.Ch. 1.1 - Prove or disprove that AB=AC implies B=C.Ch. 1.1 - 38. Prove or disprove that .
Ch. 1.1 - Prove or disprove that (AB)=(A)(B).Ch. 1.1 - 40. Prove or disprove that .
Ch. 1.1 - Express (AB)(AB) in terms of unions and...Ch. 1.1 - 42. Let the operation of addition be defined on...Ch. 1.1 - 43. Let the operation of addition be as defined in...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - True or False
Label each of the following...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - 1. For the given sets, form the Cartesian...Ch. 1.2 - For each of the following mapping, state the...Ch. 1.2 - 3. For each of the following mappings, write out ...Ch. 1.2 - For each of the following mappings f:ZZ, determine...Ch. 1.2 - 5. For each of the following mappings, determine...Ch. 1.2 - 6. For the given subsets and of Z, let and...Ch. 1.2 - 7. For the given subsets and of Z, let and...Ch. 1.2 - 8. For the given subsets and of Z, let and...Ch. 1.2 - For the given subsets A and B of Z, let f(x)=2x...Ch. 1.2 - For each of the following parts, give an example...Ch. 1.2 - For the given f:ZZ, decide whether f is onto and...Ch. 1.2 - 12. Let and . For the given , decide whether is...Ch. 1.2 - 13. For the given decide whether is onto and...Ch. 1.2 - 14. Let be given by
a. Prove or disprove that ...Ch. 1.2 - 15. a. Show that the mapping given in Example 2...Ch. 1.2 - 16. Let be given by
a. For , find and .
b. ...Ch. 1.2 - 17. Let be given by
a. For find and.
b. For...Ch. 1.2 - 18. Let and be defined as follows. In each case,...Ch. 1.2 - Let f and g be defined in the various parts of...Ch. 1.2 - In Exercises 20-22, Suppose and are positive...Ch. 1.2 - In Exercises 20-22, Suppose and are positive...Ch. 1.2 - In Exercises 20-22, Suppose and are positive...Ch. 1.2 - Let a and b be constant integers with a0, and let...Ch. 1.2 - 24. Let, where and are nonempty.
Prove that for...Ch. 1.2 - 25. Let, where and are non empty, and let and ...Ch. 1.2 - 26. Let and. Prove that for any subset of T of...Ch. 1.2 - 27. Let , where and are nonempty. Prove that ...Ch. 1.2 - 28. Let where and are nonempty. Prove that ...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - True or False
Label each of the following...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - For each of the following pairs and decide...Ch. 1.3 - For each pair given in Exercise 1, decide whether ...Ch. 1.3 - Let . Find mappings and such that.
Ch. 1.3 - Give an example of mappings and such that one of...Ch. 1.3 - Give an example of mapping and different from...Ch. 1.3 - 6. a. Give an example of mappings and , different...Ch. 1.3 - 7. a. Give an example of mappings and , where is...Ch. 1.3 - Suppose f,g and h are all mappings of a set A into...Ch. 1.3 - Find mappings f,g and h of a set A into itself...Ch. 1.3 - Let g:AB and f:BC. Prove that f is onto if fg is...Ch. 1.3 - 11. Let and . Prove that is one-to-one if is...Ch. 1.3 - Let f:AB and g:BA. Prove that f is one-to-one and...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - Label each of the following statements as either...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False Label each of the following...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False Label each of the following...Ch. 1.4 - 1. Decide whether the given set is closed with...Ch. 1.4 - In each part following, a rule that determines a...Ch. 1.4 - 3. Let be a set of three elements given by . In...Ch. 1.4 - 4. Let be a set of three elements given by . In...Ch. 1.4 - 5. Let be the set of four elements given by with...Ch. 1.4 - Let S be the set of four elements given by S={...Ch. 1.4 - 7. Prove or disprove that the set of nonzero...Ch. 1.4 - 8. Prove or disprove that the set of all odd...Ch. 1.4 - 9. The definition of an even integer was stated in...Ch. 1.4 - 10. Prove or disprove that the set of all nonzero...Ch. 1.4 - Prove or disprove that the set B={ z3|zZ } is...Ch. 1.4 - 12. Prove or disprove that the set of non zero...Ch. 1.4 - Assume that is an associative binary operation on...Ch. 1.4 - Assume that is a binary operation on a non empty...Ch. 1.4 - 15. Let be a binary operation on the non empty...Ch. 1.4 - Assume that is an associative binary operation on...Ch. 1.5 - True or False Label each of the following...Ch. 1.5 - True or False Label each of the following...Ch. 1.5 - True or False
Label each of the following...Ch. 1.5 - For each of the following mappings exhibit a...Ch. 1.5 - 2. For each of the mappings given in Exercise 1,...Ch. 1.5 - 3. If is a positive integer and the set has ...Ch. 1.5 - 4. Let , where is nonempty. Prove that a has...Ch. 1.5 - Let f:AA, where A is nonempty. Prove that f a has...Ch. 1.5 - 6. Prove that if is a permutation on , then is a...Ch. 1.5 - Prove that if f is a permutation on A, then...Ch. 1.5 - 8. a. Prove that the set of all onto mappings from...Ch. 1.5 - Let f and g be permutations on A. Prove that...Ch. 1.5 - 10. Let and be mappings from to. Prove that if is...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - True or False
Label each of the following...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Write out the matrix that matches the given...Ch. 1.6 - 2. Perform the indicated operations, if...Ch. 1.6 - 3. Perform the following multiplications, if...Ch. 1.6 - Let A=[aij]23 where aij=i+j, and let B=[bij]34...Ch. 1.6 - 5. Show that the matrix equation is equivalent to...Ch. 1.6 - 6. Write a single matrix equation of the form ...Ch. 1.6 - Let ij denote the Kronecker delta: ij=1 if i=j,...Ch. 1.6 - Let S be the set of four matrices S={I,A,B,C},...Ch. 1.6 - 9. Find two square matrices and such that.
Ch. 1.6 - Find two nonzero matrices A and B such that AB=BA.Ch. 1.6 - 11. Find two nonzero matrices and such that.
Ch. 1.6 - 12. Positive integral powers of a square matrix...Ch. 1.6 - For the matrices in Exercise 12, evaluate (A+B)2...Ch. 1.6 - Assume that A1 exists and find a solution X to...Ch. 1.6 - 15. Assume that are in and with and invertible....Ch. 1.6 - a. Prove part d of Theorem 1.30. b. Prove part e...Ch. 1.6 - a. Prove part a. of Theorem 1.34. b. Prove part b....Ch. 1.6 - Prove part b of Theorem 1.35.
Theorem 1.35 ...Ch. 1.6 - Let a and b be real numbers and A and B elements...Ch. 1.6 - Prove that if then.
Ch. 1.6 - Suppose that A is an invertible matrix over and O...Ch. 1.6 - Let be the set of all elements of that have one...Ch. 1.6 - Prove that the set S={[abba]|a,b} is closed with...Ch. 1.6 - Prove or disprove that the set of diagonal...Ch. 1.6 - Let A and B be square matrices of order n over...Ch. 1.6 - Let and be square matrices of order over ....Ch. 1.6 - A square matrix A=[aij]n with aij=0 for all ij is...Ch. 1.6 - Let a,b,c,andd be real numbers. If adbc0, show...Ch. 1.6 - Let A=[abcd] over . Prove that if adbc=0, then A...Ch. 1.6 - Let be elements of where is not a zero matrix....Ch. 1.6 - Let A,BandC be square matrices of order n over ....Ch. 1.6 - Let A and B be nn matrices over such that A1 and...Ch. 1.7 - Label each of the following statements as either...Ch. 1.7 - True or False
Label each of the following...Ch. 1.7 -
True or False
Label each of the following...Ch. 1.7 - Label each of the following statements as either...Ch. 1.7 - True or False
Label each of the following...Ch. 1.7 - Label each of the following statements as either...Ch. 1.7 - For determine which of the following relations...Ch. 1.7 - 2. In each of the following parts, a relation is...Ch. 1.7 - a. Let R be the equivalence relation defined on Z...Ch. 1.7 - 4. Let be the relation “congruence modulo 5”...Ch. 1.7 - 5. Let be the relation “congruence modulo ”...Ch. 1.7 - In Exercises 610, a relation R is defined on the...Ch. 1.7 - In Exercises 610, a relation R is defined on the...Ch. 1.7 - In Exercises 610, a relation R is defined on the...Ch. 1.7 - In Exercises 610, a relation R is defined on the...Ch. 1.7 - In Exercises , a relation is defined on the set ...Ch. 1.7 - Let be a relation defined on the set of all...Ch. 1.7 - Let and be lines in a plane. Decide in each case...Ch. 1.7 - 13. Consider the set of all nonempty subsets of ....Ch. 1.7 - In each of the following parts, a relation is...Ch. 1.7 - Let A=R0, the set of all nonzero real numbers, and...Ch. 1.7 - 16. Let and define on by if and only if ....Ch. 1.7 - In each of the following parts, a relation R is...Ch. 1.7 - Let (A) be the power set of the nonempty set A,...Ch. 1.7 - For each of the following relations R defined on...Ch. 1.7 - Give an example of a relation R on a nonempty set...Ch. 1.7 - 21. A relation on a nonempty set is called...Ch. 1.7 - A relation R on a nonempty set A is called...Ch. 1.7 - A relation R on a nonempty set A is called...Ch. 1.7 - For any relation on the nonempty set, the inverse...Ch. 1.7 - 25. Let , , and . Write out and .
Ch. 1.7 - 26. Let , , and . Write out and .
Ch. 1.7 - Prove Theorem 1.40: If is an equivalence relation...Ch. 1.7 - 28. Let , and . Define the relation R on A by if...Ch. 1.7 - 29. Suppose , , represents a partition of the...Ch. 1.7 - Suppose thatis an onto mapping from to. Prove that...

Find more solutions based on key concepts

Show solutions For Problems 14-20, evaluate each algebraic expression for the given values of the variables. 7(x2)+6(x1)4(x+3)...

Intermediate Algebra

Use the map in Illustration 1 to find the approximate air distance between each pair of cities: ILLUSTRATION 1 ...

Elementary Technical Mathematics

Simplify the expressions in Exercises 97106. (xy)1/3(yx)1/3

Applied Calculus

SOC The price of a gallon of regular gas for 20 nations is reported below. Compute the mean, median, range, int...

Essentials Of Statistics

A research study reports that alcohol consumption is significantly higher for students at a state university th...

Statistics for The Behavioral Sciences (MindTap Course List)

An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, 8, or 9, in succession. a....

Probability and Statistics for Engineering and the Sciences

What do we mean when we say a histogram is skewed to the left? to the right? What is a bimodal histogram? Discu...

Understanding Basic Statistics

Write each complex number in trigonometric form, once using degrees and once using radians. In each case, begin...

Trigonometry (MindTap Course List)

Solve each equation in Exercises 107120 for x, rounding your answer to four significant digits where necessary....

Finite Mathematics and Applied Calculus (MindTap Course List)

Gaucher, Friesen, and Kay (2011) found that words they identified as masculine themed (such tut competitive, in...

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or gi...

Single Variable Calculus

Convert the expressions in Exercises 8596 radical form. 0.2x2/3+37x1/2

Finite Mathematics

Form of the Partial Fraction Decomposition Write the form of the partial fraction decomposition of the function...

Precalculus: Mathematics for Calculus (Standalone Book)

Evaluate the definite integrals in Problems 1-32.

Mathematical Applications for the Management, Life, and Social Sciences

Graph several members of the family of functions f(x)=11+aebx where a 0. How does the graph change when b cha...

Single Variable Calculus: Early Transcendentals, Volume I

Use Table 12-1 to calculate the future value of the following annuities due.
Annuity Payment Time Nominal Inter...

Contemporary Mathematics for Business & Consumers

In Exercises 51-58, use the Vertical Line Test to determine whether the graph represents y as a function of x. ...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Use the result of Exercise 74 to prove the following: a arctan12+arctan13=/4 b 2arctan13+arctan17=/4

Calculus (MindTap Course List)

Use the recursive definition for Fibonacci numbers and deductive reasoning to verify that, for Fibonacci number...

Mathematical Excursions (MindTap Course List)

Test the series for convergence or divergence. 25. n=1n!en2

Multivariable Calculus

Graphing Linear Equations in Exercises 17-26, sketch the graph of the linear equation. Use a graphing utility t...

Calculus: An Applied Approach (MindTap Course List)

Finding the Center of Mass In Exercises 7-10, find the mass and center of mass of the lamina for each density. ...

Calculus: Early Transcendental Functions

Tangent lines T1, and T2, are drawn at two points P1, and P2, on the parabola y = x2 and they intersect at a po...

Calculus: Early Transcendentals

If angles ABC and DEF were measured with a protractor, what does intuition tell you about the degree measures o...

Elementary Geometry For College Students, 7e

Differentiate the function. y = ln(ex + xex)

Single Variable Calculus: Early Transcendentals

WRITING ABOUT CONCEPTS Finding Functions Find differentiable functions f and g such that limxf(x)=limxg(x)=andl...

Calculus: Early Transcendental Functions (MindTap Course List)

Evaluating a Limit In Exercises 15-42, evaluate the limit, using LHpitals Rule if necessary. limx1x111x41

Calculus of a Single Variable

SOCIAL MEDIA ACCOUNTS In a survey of 1000 social media account holders, the following questions were asked: How...

Finite Mathematics for the Managerial, Life, and Social Sciences

In exercises 23-26, find the exact areas of shaded region. Square inscribed in a circle

Elementary Geometry for College Students

A study, described in the paper Prediction of Defibrillation Success from a Single Defibrillation Threshold Mea...

Introduction To Statistics And Data Analysis

The graph of f(x) = 3x5 − 5x3 is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

is continuous for all (x, y) such that:
x ≥ 0 and y ≥ 0
y > x
y > –x
–y ≤ x

Study Guide for Stewart's Multivariable Calculus, 8th

Solve each inequality and express the solution using interval notation. |x+2||x+1|

College Algebra (MindTap Course List)

City Trucking, Inc., claims consistent delivery times for its routine customer deliveries. A sample of 22 truck...

Statistics for Business & Economics, Revised (MindTap Course List)

Identify and explain the common threats to external validity and identify threats when they appear in a researc...

Research Methods for the Behavioral Sciences (MindTap Course List)

EXPLORING CONCEPTS Using a Derivative The three components of the derivative of the vector-valued function u ar...

Multivariable Calculus

Actuaries are analysts who specialize in the mathematics of risk. Actuaries often work for insurance companies ...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Identify the advantages of a multiple-group design compared to an experiment with only two groups.

Research Methods for the Behavioral Sciences (MindTap Course List)

Express the following decimal fractions as common fractions. Reduce to lowest terms. 45. 0.8717

Mathematics For Machine Technology

HOW DO YOU SEE IT? The vertices of a triangle in space are (x1,y1,z1),(x2,y2,z2), and (x3,y3,y3). Explain how t...

Calculus (MindTap Course List)

Graph Rate of Change What can be said about the graph of the dfdx in each of the following situations? a. The g...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

2. A decision maker faced with four decision alternatives and four states of nature develops the following prof...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

3. In a regression analysis involving 30 observations, the following estimated regression equation was obtained...

Essentials Of Statistics For Business & Economics

Given a positive integer n written in decimal form, the alternating sum of the digits of n is obtained by start...

Discrete Mathematics With Applications

In Exercises 17, use a truth table to show that the given argument form listed in Figure 1.70 is valid. Additio...

Mathematics: A Practical Odyssey

In Problems 3544 use the Laplace transform to solve the given initial-value problem. 39. y + 5y + 4y = 0, y(0) ...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

Sketching the Graph of a Rational Function In Exercises 15-44, (a) state the domain of the function, (b) identi...

College Algebra

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 42...

Calculus Volume 1