Chapter 8.3, Problem 28ES

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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 dasses of each relation.I is the relation defined on R as follows: For every                     m / n                     ⇔     x − y         is     an integer .

To determine

To prove:

The given relation is an equivalence relation and also find the distinct equivalence classes of given relation.

Explanation

Given information:

I is the relation defined on as follows:

For all x, y ∈ R, x I y ⇔ x - y is an integer.

Calculation:

I={(m,n)×|mn is an integer}

To prove: R is an equivalent relation.

Reflexive:

Let x

xx=0

xx is an integer, since 0 is an integer and thus (x,x)I

Symmetric:

Let us assume that (x,y)I.

By the definition of I :

xy is an integer.

However the negation of an integer is also an integer.

(xy) is an integer.

Use distributive property,

yx is an integer.

By the definition of I :

(y,x)I

Since (x,y)I implies (y,x)II is symmetric.

Transitive:

Let us assume that (x,y)I and (y,z)I.

By the definition of I :

(xy) is an integer

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