Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Textbook Question
Chapter 1.1, Problem 1TFE
True or False
Label each of the following statements as either true or false.
Two sets are equal if and only if they contain exactly the same elements.
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Question 1. Solve the system
-
x1 x2 + 3x3 + 2x4
-x1 + x22x3 + x4
2x12x2+7x3+7x4
Question 2. Consider the system
= 1
=-2
= 1
3x1 - x2 + ax3
= 1
x1 + 3x2 + 2x3
x12x2+2x3
= -b
= 4
1 For what values of a, b will the system be inconsistent?
2 For what values of a, b will the system have only one solution?
For what values of a, b will the saystem have infinitely many solutions?
Chapter 1 Solutions
Elements Of Modern Algebra
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 - Prob. 5TFECh. 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 - Prob. 1ECh. 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 - Prob. 5ECh. 1.1 - 6. Determine whether each of the following is...Ch. 1.1 - Prob. 7ECh. 1.1 - 8. Describe two partitions of each of the...Ch. 1.1 - Prob. 9ECh. 1.1 - Prob. 10ECh. 1.1 - Prob. 11ECh. 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 - Prob. 14ECh. 1.1 - Prob. 15ECh. 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 - Prob. 19ECh. 1.1 - In Exercises 1435, prove each statement. (AB)=ABCh. 1.1 - Prob. 21ECh. 1.1 - Prob. 22ECh. 1.1 - In Exercises 14-35, prove each statement.
23.
Ch. 1.1 - Prob. 24ECh. 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 - Prob. 28ECh. 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 - Prob. 39ECh. 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 - Prob. 1ECh. 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 - Prob. 19ECh. 1.2 - Prob. 20ECh. 1.2 - In Exercises 20-22, Suppose and are positive...Ch. 1.2 - Prob. 22ECh. 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 - Prob. 1ECh. 1.4 - In each part following, a rule that determines a...Ch. 1.4 - Prob. 3ECh. 1.4 - Prob. 4ECh. 1.4 - Prob. 5ECh. 1.4 - Prob. 6ECh. 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 - Prob. 11ECh. 1.4 - Prob. 12ECh. 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 - Prob. 3TFECh. 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 - Prob. 3ECh. 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 - Prob. 3TFECh. 1.6 - Prob. 4TFECh. 1.6 - Prob. 5TFECh. 1.6 - Prob. 6TFECh. 1.6 - Prob. 7TFECh. 1.6 - Prob. 8TFECh. 1.6 - Prob. 9TFECh. 1.6 - Prob. 10TFECh. 1.6 - Prob. 11TFECh. 1.6 - Label each of the following statements as either...Ch. 1.6 - Write out the matrix that matches the given...Ch. 1.6 - Prob. 2ECh. 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 - Prob. 5ECh. 1.6 - Prob. 6ECh. 1.6 - Let ij denote the Kronecker delta: ij=1 if i=j,...Ch. 1.6 - Prob. 8ECh. 1.6 - Prob. 9ECh. 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 - Prob. 13ECh. 1.6 - Prob. 14ECh. 1.6 - 15. Assume that are in and with and invertible....Ch. 1.6 - Prob. 16ECh. 1.6 - Prob. 17ECh. 1.6 - Prove part b of Theorem 1.35.
Theorem 1.35 ...Ch. 1.6 - Prob. 19ECh. 1.6 - Prob. 20ECh. 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 - Prob. 24ECh. 1.6 - Let A and B be square matrices of order n over...Ch. 1.6 - Prob. 26ECh. 1.6 - A square matrix A=[aij]n with aij=0 for all ij is...Ch. 1.6 - Prob. 28ECh. 1.6 - Prob. 29ECh. 1.6 - Prob. 30ECh. 1.6 - Prob. 31ECh. 1.6 - Prob. 32ECh. 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 - Prob. 23ECh. 1.7 - For any relation on the nonempty set, the inverse...Ch. 1.7 - Prob. 25ECh. 1.7 - Prob. 26ECh. 1.7 - Prove Theorem 1.40: If is an equivalence relation...Ch. 1.7 - Prob. 28ECh. 1.7 - 29. Suppose , , represents a partition of the...Ch. 1.7 - Suppose thatis an onto mapping from to. Prove that...
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