   Chapter 8.3, Problem 25ES ### 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 dasses of each relation.A is the “absolute value” relation defined on R as follows: For   every  x , y ∈ A y       ⇔ | x | = | 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:

A is the “absolute value” relation defined on R as follows:

For all x, y ∈ R, x A y ⇔ | x | = | y|.

Proof:

A={(x,y)R×R||x|=|y|}

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

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

Reflexive:

Let xR

Since |x|=|x|,(x,x)A.

Since (x,x)A for all xR,A is indeed reflexive

Symmetric:

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

|x|=|y|

However, if |x|=|y|, then |y|=|x| also holds.

(y,x)A

Since (x,y)A implies (y,x)AA is symmetric.

Transitive:

Let us assume that (x,y)A and (y,z)A

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