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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?

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?

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 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...
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please send handwritten solution for Q 1 part b
Q1 SHORT ANSWER QUESTION. No justification needed. a) Define the equivalence relation S on the integers by Sy + 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 € [3] n [5], then how are [3] and [5] related?
Transcribed Image Text:Q1 SHORT ANSWER QUESTION. No justification needed. a) Define the equivalence relation S on the integers by Sy + 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 € [3] n [5], then how are [3] and [5] related?
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