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xSy + 11|(x – y). State the least positive integer in the equivalence class of [14]. b) For a set X = {1, 2, 3, 4}, let P be the partition

xSy + 11|(x – y). State the least positive integer in the equivalence class of [14]. b) For a set X = {1, 2, 3, 4}, let P be the partition

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 8E: In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R...
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a) Define the equivalence relation S on the integers by xSy 11|(x – y). State the least positive integer in the equivalence class of [14]. b) For a set X = {1,2, 3, 4}, let P be the partition {{1,3}, {2,4}}. The partition P induces a relation R on X. How many distinct ordered pairs are there in R? c) For a set X = {1, 2, 3, 4, 5}, let P be the partition {{1,3, 4}, {2}, {5}}. In the equivalence relation on X induced by P, how many distinct equivalence classes are there? d) Let X = {1, 2, 3, 4}. Define a binary relation R on subsets of X by ARB if and only if A and B have the same number of elements. Does {1}R{1, 2} hold? Answer yes or no. e) Suppose that S is an equivalence relation on a set A = {1,2, 3, 4, 5, 6, 7}. If 4 E [3] n [5], then how are |3| [3] and [5] related?
Transcribed Image Text:a) Define the equivalence relation S on the integers by xSy 11|(x – y). State the least positive integer in the equivalence class of [14]. b) For a set X = {1,2, 3, 4}, let P be the partition {{1,3}, {2,4}}. The partition P induces a relation R on X. How many distinct ordered pairs are there in R? c) For a set X = {1, 2, 3, 4, 5}, let P be the partition {{1,3, 4}, {2}, {5}}. In the equivalence relation on X induced by P, how many distinct equivalence classes are there? d) Let X = {1, 2, 3, 4}. Define a binary relation R on subsets of X by ARB if and only if A and B have the same number of elements. Does {1}R{1, 2} hold? Answer yes or no. e) Suppose that S is an equivalence relation on a set A = {1,2, 3, 4, 5, 6, 7}. If 4 E [3] n [5], then how are |3| [3] and [5] related?
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