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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Suppose that R is a partial order relation on a set A and that B is a sunset of A. The restriction of R to B is defined as follows:

The restriction of R to B

   = { ( x , y ) | x B , y B , and ( x , y ) R } .

In other words, two elements of B are related by the restriction of R to B if, and only if, they are related by R. Prose that the restriction of R to B is a partial order relation on B. (In less formal language, this says that a subset of a partially ordered set is partially ordered.)

To determine

To prove:

The restriction of R to B is a partial order relation on B.

Explanation

Given information:

The restriction of R to B is R1={(x,y)/xB,yB,(x,y)R}.

Concept used:

A relation is called partial order relation if and only if the relation is reflexive, antisymmetric, and transitive.

Proof:

Consider the set A and R be the partial order relation.

BA

Let R1 be the restriction of R to B.

The restriction of R to B is R1={(x,y)/xB,yB,(x,y)R}.

The objective is to prove that R1 is a partial order relation on B.

A relation is called partial order relation if and only if the relation is reflexive, antisymmetric, and transitive,

To prove that R1 is reflexive.

For xB,xB,(x,x)R1

xR1x

This shows that R1 is reflexive.

However, R1 is a restriction of R to B.

So, R1 is reflexive on B.

To prove that R1 is antisymmetric.

For xB,yB(x,y)R1

xR1y

And yB,xB,(y,x)R1

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