Elements Of Modern Algebra

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Prove A is a subset of (A U B)

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Prove that if f is a permutation on A, then (f1)1=f.

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Label each of the following statements as either true or false. Every upper bound of a nonempty set S must be an element of S.

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15. Let be a binary operation on the non empty set . Prove that if contains an identity element with respect to , the identity element is unique.

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9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to
a. addition defined on .
b. multiplication defined on .

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Label each of the following statements as either true or false. The least upper bound of a nonempty set S is unique.

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2. Prove that is commutative if and only if is commutative.

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In Exercises 13-24, prove the statements concerning the relation on the set of all integers.
19. If and, then.

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43. Let the operation of addition be as defined in Exercise 42. Prove each of the following statements.
a. b.
42. Let the operation of addition be defined on subsets and of by

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Label each of the following statements as either true or false.
2. for all nonempty sets A and B.

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In Exercises , prove the statements concerning the relation on the set of all integers.
14. If and , then .

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In Exercises , prove the statements concerning the relation on the set of all integers.
17. If and , then .

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Suppose thatis an onto mapping from to. Prove that if ℒ, is a partition of, then ℒ, is a partition of.

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