   Chapter 8.2, Problem 22ES ### 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 9-33, determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answer. Let X = { a , b , c } and P ( X ) be the power set of X. A relation N is defined on P ( X ) as follows: For every A , B , ∈ P ( X ) ,   A N B   ⇔ the number of elements in A is not equal to the number of elements in B.

To determine

To justify whether the given relation is reflexive, symmetric, transitive, or none of these.

Explanation

Given information:

Let X = { a, b, c } and P(X) be the power set of X. A relation N Is defined on P(X) as follows: For all A, B ∈ P(X), A N B ⇔ the number of elements in A is not equal to the number of elements in B.

Calculation:

Consider the given set X={a,b,c}

Let us consider that A=P(X)

The relation N Is given by

N={B,CP(X)|B has a different number of elements than C}

Reflexive:

The relation N is reflective if (B,B)N for every element BA.

Since A=P(X),

N is reflexive if it contains (B,B) for all BP(X).

We note that N does not contain (B,B) because a set never a different number of elements than itself and thus N is not reflexive.

It is not reflexive

Symmetric:

The relation N on a set A is symmetric if (C,B)N whenever (B,C)N

Let us assume that (B,C)N, by definition of N :

B has a different number of elements than C

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