   Chapter 8.5, Problem 15ES ### 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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# Suppose a relation R on a set A is reflexive, symmetric, transitive, and antisymmetric. What can you conclude about R? Prove your answer.

To determine

To find:

The conclusion about R if it is reflexive, symmetric, transitive, and antisymmetric.

Explanation

Given information:

R and S are antisymmetric

Concept used:

Let R be a relation on a set A. Then R is said to be anti-symmetric relation if and only if xRy and yRx implies x=y for every x,yA.

Calculation:

A relation R on a set A is such that it is reflexive, symmetric, transitive, and antisymmetric.

The objective is to find the kind of relation R is.

Reflexive: Let R be a reltion on a set A. Then R is reflexive if, and only if, for all xA,xRx

or (x,x)R.

Symmetric: Let R be a relation on a set A. Then R is symmetric if, and only if, for all x,yA, if xRy(or (x,y)R) then yRx(or (y,x)R).

Transitive: let R be a relation on a set A .Then R is transitive if, and only if, for all x,y,zA, if xRy and yRz, then xRz. That is, if (x,y)R and (y,z)R then (x,z)R.

Antisymmetric: let R be a relation on a set A. Then R is antisymmetric if, and only if, for all x,yA, if xRy and yRx, then x=y

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