   Chapter 1.7, Problem 29E

Chapter
Section
Textbook Problem
1 views

# Suppose { A λ } , λ ∈ £ , represents a partition of the nonempty set A. Define R on A by x R y if and only if there is a subset A λ such that x ∈ A λ   and   y ∈ A λ . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets A λ .

To determine

To prove: R is an equivalence relation on A and that the equivalence classes of R are the subsets Aλ.

Explanation

Given Information:

Suppose {Aλ}, λ£, represents a partition of the nonempty set A. R is defined on A by xRy if and only if there is a subset Aλ such that xAλandyAλ.

Proof:

A relation R on a nonempty set A is an equivalence relation if the following conditions are satisfied for arbitrary x,y,z in A:

1. Reflexive Property: xRxforallxA.

2. Symmetric Property: If xRy, then yRx.

3. Transitive Property: If xRy and yRz, then xRz.

Consider the given information.

Suppose {Aλ}, λ£, represents a partition of the nonempty set A. R is defined on A by xRy if and only if there is a subset Aλ such that xAλandyAλ.

1. xRx, since xAxλ£Aλ.

2. xRyx,yAλforsomeλy,xAλforsomeλyRx

3

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Convert from radians to degrees. 9. 512

Single Variable Calculus: Early Transcendentals, Volume I

#### Solve the equations in Exercises 126. x1x=0

Finite Mathematics and Applied Calculus (MindTap Course List)

#### True or False:

Study Guide for Stewart's Multivariable Calculus, 8th 