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Math Algebra Elements Of Modern Algebra

In 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 .

In 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 .

Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 1.7, Problem 2E

In 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 .

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Chapter 1.7, Problem 1E

Chapter 1.7, Problem 3E

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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. 5TFE 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 - Prob. 1E 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 - Prob. 5E Ch. 1.1 - 6. Determine whether each of the following is...Ch. 1.1 - Prob. 7E Ch. 1.1 - 8. Describe two partitions of each of the...Ch. 1.1 - Prob. 9E Ch. 1.1 - Prob. 10E Ch. 1.1 - Prob. 11E 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 - Prob. 14E Ch. 1.1 - Prob. 15E Ch. 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. 19E Ch. 1.1 - In Exercises 1435, prove each statement. (AB)=AB Ch. 1.1 - Prob. 21E Ch. 1.1 - Prob. 22E Ch. 1.1 - In Exercises 14-35, prove each statement. 23. Ch. 1.1 - Prob. 24E 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 - Prob. 28E 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 - Prob. 39E 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 - Prob. 1E 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 - Prob. 19E Ch. 1.2 - Prob. 20E Ch. 1.2 - In Exercises 20-22, Suppose and are positive...Ch. 1.2 - Prob. 22E 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 - Prob. 1E Ch. 1.4 - In each part following, a rule that determines a...Ch. 1.4 - Prob. 3E Ch. 1.4 - Prob. 4E Ch. 1.4 - Prob. 5E Ch. 1.4 - Prob. 6E 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 - Prob. 11E Ch. 1.4 - Prob. 12E 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 - Prob. 3TFE 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 - Prob. 3E 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 - Prob. 3TFE Ch. 1.6 - Prob. 4TFE Ch. 1.6 - Prob. 5TFE Ch. 1.6 - Prob. 6TFE Ch. 1.6 - Prob. 7TFE Ch. 1.6 - Prob. 8TFE Ch. 1.6 - Prob. 9TFE Ch. 1.6 - Prob. 10TFE Ch. 1.6 - Prob. 11TFE 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 - Prob. 2E 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 - Prob. 5E Ch. 1.6 - Prob. 6E Ch. 1.6 - Let ij denote the Kronecker delta: ij=1 if i=j,...Ch. 1.6 - Prob. 8E Ch. 1.6 - Prob. 9E 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 - Prob. 13E Ch. 1.6 - Prob. 14E Ch. 1.6 - 15. Assume that are in and with and invertible....Ch. 1.6 - Prob. 16E Ch. 1.6 - Prob. 17E Ch. 1.6 - Prove part b of Theorem 1.35. Theorem 1.35 ...Ch. 1.6 - Prob. 19E Ch. 1.6 - Prob. 20E 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 - Prob. 24E Ch. 1.6 - Let A and B be square matrices of order n over...Ch. 1.6 - Prob. 26E Ch. 1.6 - A square matrix A=[aij]n with aij=0 for all ij is...Ch. 1.6 - Prob. 28E Ch. 1.6 - Prob. 29E Ch. 1.6 - Prob. 30E Ch. 1.6 - Prob. 31E Ch. 1.6 - Prob. 32E 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 - Prob. 23E Ch. 1.7 - For any relation on the nonempty set, the inverse...Ch. 1.7 - Prob. 25E Ch. 1.7 - Prob. 26E Ch. 1.7 - Prove Theorem 1.40: If is an equivalence relation...Ch. 1.7 - Prob. 28E Ch. 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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