   Chapter 8.3, Problem 29ES ### 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.Define P on the R × R of order paors of real numbers as follows: For every ( w , x ) , ( y , z ) ∈ R × R , ( w , x ) P ( y , z ) ⇔ w = y .

To determine

To prove:

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

Explanation

Given information:

Define P on the set × of ordered pairs of real numbers as follows:

For all (w, x),(y, z)× ,

(w, x)P(y, z)w=y.

Calculation:

P={(w,x),(y,z)×|w=y}

To prove: P is an equivalent relation.

Reflexive:

Let (x,y)×

Since x=x, so (x,y)P(x,y)

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

Symmetric:

Let us assume that (w, x)P(y, z)

By the definition of P :

w=y

However, w = y is equivalent with y = w.

So, (y, z)P(w, x)

Since (w, x)P(y, z) implies (y, z)P(w, x)P is symmetric

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