   Chapter 8.5, Problem 14ES ### 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 A = { a , b , c } . Describe all partial order relations on A for which a is a maximal element. Describe all order relations on A for which a is a minimal element.

To determine

(a)

To find:

All partial order relations on A for which a is the maximal element.

Explanation

Given information:

A={a,b,c}

Concept used:

If given set is reflexive, antisymmetric and transitive then the given set is called of partial order.

Calculation:

Consider a set A={a,b,c}.

Objective is to describe all partial order relations on a set A={a,b,c} for a is a maximal element.

A relation R on the set A is a partial order relation if R is reflexive, antisymmetric, and transitive.

The partial order relation on A={a,b,c} for which a is a maximal element is as follows.

R1={(a,a)(b,b)(c,c)}

Since (a,a)(b,b)(c,c) follows reflexive, antisymmetric, and transitive.

aa,bb,cc Which is the maximal property.

So, R1 is partial order relation.

Second partial order relations is as follows

R2={(a,a),(b,b),(c,c),(b,a)}

Here (a,a),(b,b),(c,c) follows partial order by R1 and ba. Which is a maximal property.

R3={(a,a),(b,b),(c,c),(c,a)}

Since a is maximal ca and (a,a),(b,b),(c,c) by R1.

Next partial order relation is as follows.

R4={(a,a),(b,b),(c,c),(b,a),(c,a)}

Since by R1,R2 and R3 are partial order so R1R2R3=R4 is also partial order.

Next partial order relation is as follows.

R5={(a,a)(b,b)(c,c)(c,b)(c,a)}

R6={(a,a)(b,b)(c,a)(b,c)(b,a)}R7={(a,a)(b,b)(c,c)(c,b)(b,a)(c,a)}R8={(a,a)(b,b)(c,c)(b,c)(b,a)(c,a)}

Since a is maximal as by R1,R5 and R6

To determine

(b)

To find:

All the partial order relations on A for which a is a minimal element.

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