   Chapter 8.5, Problem 5ES ### 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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# Let R be the set of all real numbers and define a relation R on R × R as follows: For every ( a , b ) and ( c , d ) in R × R . ( a , b ) R ( c , d )     ⇔     either a < c     or both  a = c   and  b ≤ d . Is R a partial order relation? Prove or give a counterexample.

To determine

To check:

Whether R is a partial order relation or not.

Explanation

Given information:

Let be the set of all real numbers and define a relation R on × as follows:

For all (a,b) and (c,d) in ×,

(a,b)R(c,d)either a<c or both a=c and bd.

Concept used:

A relation that is reflexive, antisymmetric, and transitive is called a partial order.

Here, given that R is a relation on × as follows.

For all (a,b) and (c,d) in ×.

(a,b)R(c,d)either a<c or both a=c and bd...........(1)

Calculation:

Verify reflexivity of the relation R as follows.

Suppose (a,b)×. Then (a,b)R(a,b) because a=a and bb.

Verify anti-symmetricity of the relation R as follows.

Suppose that (a,b),(c,d)× such that (a,b)R(c,d) and (c,d)R(a,b).

Then from the definition (1), obtain the following.

“Either a<c or both a=c and bd

And either c<a or both c=a and db ”.

From a<c or a=c, conclude that ac.

From a>c or a=c, conclude that ac.

As ac and ac, this implies that a=c.

On similarly arguments, obtain the following.

bd and db,b=d

From a=c and b=d

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