   Chapter 8.3, Problem 23ES ### 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.Let P be a set of parts shipped to a company from various suppliers, S is the relation defined on P as follows: For every x , y ∈ P . x S y       ⇔   x     has   the sam part number and is shipped from the same supplier as  y .

To determine

To prove:

(1) The given relation is an equivalence relation,

(2) Describe the distinct equivalence classes of each relation.

Explanation

Given information:

Let P be a set of parts shipped to a company from various suppliers.

S is the relation defined on P as follows: For all x, y ∈ P ,

x S y ⇔ x has the same part number and is shipped from the same supplier as y.

Proof:

P=Set of all parts shipped to a company by multiple suppliersS={(x,y)P×P|x has the same part number and suppliers as y}

(1). To prove: S is an equivalent relation

A relation will be equivalence relation if it is reflexive, symmetric and transitive.

Reflexive:

Let xP

Since part x always has the same part number and supplier than itself, (x,x)S.

Since (x,x)S for all xP,S is indeed reflexive

Symmetric:

Let us assume that (x,y)S. by the definition of S :

x and y have the same part number and supplier

However, if x and y have the same part number and supplier, then y and x also have the same part number and supplier

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