Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following: (a) For each a. bE A, a × b if and only if [a] n b) = 0. (b) For each a. be A, if [a] # [b], then [a] n [2] = 0.
Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following: (a) For each a. bE A, a × b if and only if [a] n b) = 0. (b) For each a. be A, if [a] # [b], then [a] n [2] = 0.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 3E: a. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the...
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11b please
![correspon-
dence (bijection) between R* and the set of all equivalence classes for
this equivalence relation.
11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove
each of the following:
(a) For each a. bE 4, ax b if and only if [a] n [b] = 0.
(b) For each a. b e A, if (a] 7 [b], then 2e] n ( = .
(c) For each a b € A, if la] N [b] 0 then la] = b|
Explorations and Activities
12. A Partition Defines an Equivalence Relation. Let 4 = fa b.c d, e} and
Tet € = {{a, b. c} {d_e}}
fa) Explain why C is a partition of A.
Define a relation s on 4as follone Fora](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68628b5-f384-4ba7-a04f-a8f64dbf4c73%2F12d5b616-4e8f-4eca-8d94-d20d6a2d48f2%2F2v00yc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:correspon-
dence (bijection) between R* and the set of all equivalence classes for
this equivalence relation.
11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove
each of the following:
(a) For each a. bE 4, ax b if and only if [a] n [b] = 0.
(b) For each a. b e A, if (a] 7 [b], then 2e] n ( = .
(c) For each a b € A, if la] N [b] 0 then la] = b|
Explorations and Activities
12. A Partition Defines an Equivalence Relation. Let 4 = fa b.c d, e} and
Tet € = {{a, b. c} {d_e}}
fa) Explain why C is a partition of A.
Define a relation s on 4as follone Fora
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