   Chapter 8.3, Problem 31ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In 19—31, (1) prove that the relation is an equivalence relation, and (2) describe the distinct equivalence classes of each relation. 31. Let P be the set of all points in the Cartesian plane except the origin. R is the relation defined on P as follows: For every p 1 and p 2 in P, p 1   R   p 2 ⇔ p 1 and p 2 lie on the same half-tine emanating from the origin.

To determine

(a)

To prove:

Show that given relation is an equivalence relation.

Explanation

Given information:

Let P be the set of all points in the Cartesian plane except the origin. R is the relation defined on P as follows:

For all p1 and p2 in P ,

p1R p2p1 and p2 lie on the same half-line emanating from the origin.

Proof:

Let

P=Set of all points in the Cartesian plane excluding the origin.R={(p,q)P×P|p and q lie on the same half-line emanating from the origin}

To prove: R is an equivalent relation

Reflexive:

Let xP

Since a point x always lie on the same half-line emanating from the origin as itself, (x,x)R.

Since (x,x)R for all xP,R is indeed reflexive.

Symmetric:

Let us assume that (x,y)R

To determine

(b)

Describe the distinct equivalence classes of given relation.

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