   Chapter 8.3, Problem 30ES ### 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.Define Q on the set R × R as follows: For every ( w , x ) , ( y , z ) ∈ R × R , ( w , x ) Q ( y , z ) ⇔ x = z .

To determine

To prove:

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

Explanation

Given information:

Define Q on the set × as follows:

For all (w, x),(y, z)×,

(w, x)Q(y, z)x=z.

Proof:

Q={(w,x),(y,z)×|x=z}

To prove: Q is an equivalence relation.

Reflexive:

Let (x,y)×

Since y=y, ((x,y),(x,y))Q

Since ((x,y),(x,y))Q for all (x,y)×, and thus Q is reflexive.

Symmetric:

Let us assume that ((w,x),(y,z))Q.

By the definition of Q :

x=z

However, x = z is equivalent with z = x.

((y,z),(w,x))Q

Since ((w,x),(y,z))Q implies ((y,z),(w,x))QQ is symmetric

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