   Chapter 1.7, Problem 15E

Chapter
Section
Textbook Problem
1 views

# Let A = R − { 0 } , the set of all nonzero real numbers, and consider the following relations on A × A . Decide in each case whether R is an equivalence relation, and justify your answers. ( a ,     b ) R ( c ,   d ) if and only if a d = b c . ( a ,     b ) R ( c ,   d ) if and only if a b = c d . ( a ,     b ) R ( c ,   d ) if and only if a 2 + b 2 = c 2 + d 2 . ( a ,     b ) R ( c ,   d ) if and only if a − b = c − d .

(a)

To determine

Whether the relation R on set A×A defined by (a,b)R(c,d) if and only if ad=bc is an equivalence relation or not.

Explanation

Given information:

A={0}, the set of all non-zero real numbers. Relation R is defined on set A×A.

The relation R defined by (a,b)R(c,d) if and only if ad=bc.

Formula used:

Definition of equivalence relation:

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

1. Reflexive property: xRx for all xA.

2. Symmetric Property: If xRy, then yRx.

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

Explanation:

1. Reflexive property:

Let (a,b)A×A

As multiplication in set of real numbers is commutative implies that ab=ba

Thus (a,b)R(a,b).

Therefore the relation R is reflexive.

2. Symmetric Property:

Let (a,b),(c,d)A×A

As multiplication in set of real numbers is commutative implies that bc=cb and ad=da.

cb=da

(c,d)R(a,b)

Therefore the relation R is symmetric

(b)

To determine

Whether the relation R on set A×A defined by (a,b)R(c,d) if and only if ab=cd is an equivalence relation or not.

(c)

To determine

Whether the relation R on set A×A defined by (a,b)R(c,d) if and only if a2+b2=c2+d2 is an equivalence relation or not.

(d)

To determine

Whether the relation R on set A×A defined by (a,b)R(c,d) if and only if ab=cd is an equivalence relation or not.

### 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

#### In Problems 21-26, graph each function. 24.

Mathematical Applications for the Management, Life, and Social Sciences

#### In Exercises 13-20, sketch a set of coordinate axes and plot each point. 15. (3, 1)

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### Find the derivative of the function. g() = cos2

Single Variable Calculus: Early Transcendentals, Volume I

#### Given that for all x, a power series for is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

#### For

Study Guide for Stewart's Multivariable Calculus, 8th 