5 5. Consider the equivalence relation ~ on Z where x ~ y means x - y is even for x, y Є Z.Suppose wЄ Z. Prove that [w] ‡ [2] ⇒ [w] = [1].6. Suppose A is the set of counties in Leinster.(i) Give an example of a relation on A that is an equivalence relation. Prove that your

Given Equivalence Relation The relation ∼ on Z is defined as:x ∼ y if and only if x - y is even.This…

Answered: 5. Consider the equivalence relation ~…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...

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Similar questions

Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.
Does the relation is in love with have a reflexive property consider one person? a symmetric property consider two people? a transitive property consider three people?
Determine whether the set S={2x+x2,8+x3,x2+x3,4+x2} spans P3.
4. Let R be a relation defined on the set of integers by a R b if a + b = 0 or a – b = 0. Determine whether R is an equivalence relation.
You just bought a new car from a dealership that sells cars no older than 2015. Consider the relationship to be the “same age.” The relation is “same age” and the set is cars made in 2015, 2016, 2017, 2018, 2019, and 2020. Prove that this relation is an equivalence relation. Describe the partition defined by the equivalence classes.
Exercise 1.5.5. (a) Why is A- A for every set A? (b) Given sets A and B, explain why A - B is equivalent to asserting B- A. (c) For three sets A, B, and C, show that A- B and B - C implies A - C. These three properties are what is meant by saying that - is an equivalence relation.
3. Let R be a relation defined on the set A = N × N where (a, b) R (c, d) if að = cd. Show R is an equivalence relation. Then, list the elements of the equivalence class [(9, 2)].
Thank you.

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5. Consider the equivalence relation ~ on Z where x ~ y means x - y is even for x, y Є Z. Suppose wЄ Z. Prove that [w] ‡ [2] ⇒ [w] = [1]. 6. Suppose A is the set of counties in Leinster. (i) Give an example of a relation on A that is an equivalence relation. Prove that your