Math

AlgebraElements Of Modern AlgebraLet f : A → B , where A and B are nonempty. Prove that f ( S 1 ∪ S 2 ) = f ( S 1 ) ∪ f ( S 2 ) for all subsets S 1 and S 2 of A . Prove that f ( S 1 ∩ S 2 ) ⊆ f ( S 1 ) ∩ f ( S 2 ) for all subsets S 1 and S 2 of A . Give an example where there are subsets S 1 and S 2 of A such that f ( S 1 ∩ S 2 ) ≠ f ( S 1 ) ∩ f ( S 2 ) . Prove that f ( S 1 ) − f ( S 2 ) ⊆ f ( S 1 − S 2 ) for all subsets S 1 and S 2 of A . Give an example where there are subsets S 1 and S 2 of A such that f ( S 1 ) − f ( S 2 ) ≠ f ( S 1 − S 2 ) .BuyFind*arrow_forward*

8th Edition

Gilbert + 2 others

Publisher: Cengage Learning,

ISBN: 9781285463230

Chapter 1.2, Problem 24E

Textbook Problem

Let

Prove that

Prove that

Give an example where there are subsets

Prove that

Give an example where there are subsets

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 37-46, evaluate each of the algebraic expressions for the given values of the variables. 5(2x3y) f...

Intermediate Algebra

Linear Equations The given equation is either linear or equivalent to a linear equation. Solve the equation. 21...

Precalculus: Mathematics for Calculus (Standalone Book)

Calculate the iterated integral. 16. 0101(x+y)2dxdy

Calculus: Early Transcendentals

Evaluate the expression sin Exercises 116. 22212424

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the first partial derivatives of the function. 30. F(,)=t3+1dt

Multivariable Calculus

Find the missing dimension in each figure in Illustration 10. ILLUSTRATION 10

Elementary Technical Mathematics

The accompanying specific gravity values for various wood types used in construction appeared in the article Bo...

Probability and Statistics for Engineering and the Sciences

For Exercises 39 to 44, make drawings as needed. Plane R is tangent to a sphere O at point T. How are radius OT...

Elementary Geometry For College Students, 7e

The range is completely determined by the two extreme scores in a distribution. The standard deviation on the, ...

Statistics for The Behavioral Sciences (MindTap Course List)

SOC A random sample of 100 cable TV movies contained an average of 3.38 acts of physical violence per program w...

Essentials Of Statistics

In Exercises 13 to 22, insert either or in the blank space between the sets to make a true statement, the set...

Mathematical Excursions (MindTap Course List)

Show that the equation represents a circle and find the center and radius. 9. 2x2 + 2y2 x + y = 1

Single Variable Calculus: Early Transcendentals, Volume I

The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the eq...

Calculus (MindTap Course List)

Convert each expression in Exercises 25-50 into its technology formula equivalent as in the table in the text. ...

Applied Calculus

In Problems 17-20, use a graphing calculator to investigate whether each of the following functions has any di...

Mathematical Applications for the Management, Life, and Social Sciences

determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give a...

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

Write each angle as a sum or difference involving . For example, 5/6=/6. 23

Trigonometry (MindTap Course List)

To convert an improper fraction to a whole or mixed number, we ___________ the numerator by the denominator. (2...

Contemporary Mathematics for Business & Consumers

Finding a Limit In Exercises 41-46, write a sampler function that agrees with the given function at all but one...

Calculus: Early Transcendental Functions

Finding Points of Intersection In Exercises 41-48, find the points of intersection (if any) of the graphs of th...

Calculus: An Applied Approach (MindTap Course List)

Convert the expressions in Exercises 8596 radical form. 94(1x)7/3

Finite Mathematics

For what values of x is g continuous? g(x)={0ifxisrationalxifxisirrational

Single Variable Calculus

A researcher is interested in the fast-food eating habits of American college students. A group of 50 students ...

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

In west Texas, water is extremely important. The following data represent pH levels in groundwater for a random...

Understanding Basic Statistics

Finding Points of Inflection In Exercises15-36, find the points of inflection and discuss the concavity of the ...

Calculus: Early Transcendental Functions (MindTap Course List)

a. What is the maximin strategy for the row player in a two-person game that is represented by a payoff matrix?...

Finite Mathematics for the Managerial, Life, and Social Sciences

Find the exact length of the curve. 20. y = 1 ex, 0 x 2

Single Variable Calculus: Early Transcendentals

Sketching the Graph of a Trigonometric Function In Exercises 55-66, sketch the graph of the function. y=2sec2x

Calculus of a Single Variable

With an initial estimate of −1, use Newton’s Method once to estimate a zero of f(x) = 5x2 + 15x + 9.

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

Cost per serving (in cents) for six high-fiber cereals rated very good and for nine high-fiber cereals rated go...

Introduction To Statistics And Data Analysis

The length of the curve given by x = 3t2 + 2, y = 2t3, is:

Study Guide for Stewart's Multivariable Calculus, 8th

Where C is a real number, the lines 3x+2y=C, 2x+y=5, and x-y=4 are concurrent. Determine the value of C.

Elementary Geometry for College Students

Practice Find the slope of the line passing through each pair of points, if possible. P(a+b,c);Q(b+c,a) assume ...

College Algebra (MindTap Course List)

Assume a binomial probability distribution has p = .60 and n = 200. a. What are the mean and standard deviation...

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

Applications A doctors office staff studied the waiting times for patients who arrive at the office with a requ...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Finding an Indefinite Integral Involving Secant and Tangent In Exercises 21-34, find the indefinite integral. s...

Calculus (MindTap Course List)

Identify the strengths and weaknesses of a multiple-baseline design and describe the circumstances in which it ...

Research Methods for the Behavioral Sciences (MindTap Course List)

Define an operational definition and explain the purpose and the limitations of operational definitions.

Research Methods for the Behavioral Sciences (MindTap Course List)

Finding Surface AreaIn Exercises 316, find the area of the surface given by z=f(x,y) that lies above the region...

Multivariable Calculus

In Exercises 17-22, show that the given sets of points are equivalent by establishing a one-to-one corresponden...

Mathematics: A Practical Odyssey

In each of the following exercises, the top, front, and right side views of a compound-angular surface are show...

Mathematics For Machine Technology

TEST YOUR UNDERSTANDING | FOR EXAMPLE 4.11 A small dog will weigh 20pounds at adulthood. The following table sh...

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

Writing Equations from Graphs In Exercises 51-56, identify the parent function and the transformation represent...

College Algebra

Ten points Libeled A. B. C. D. E. F. G. H, I. J axe arranged in a plane in such a way that no three Me on the s...

Discrete Mathematics With Applications

Top Rated Television Show Networks. Nielsen Media Research tracks the top-rated television shows. The following...

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

The Shroud of Turin, which shows the negative image of the body of a man who appears to have been crucified, is...

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

CEO Tenure. A shareholders’ group, in lodging a protest, claimed that the mean tenure for a chief executive off...

Essentials Of Statistics For Business & Economics

For the following exercises, determine whether the statement is true or false. Explain why. 98. f(x)=(4x+1)/(7x...

Calculus Volume 1