   Chapter 8.2, Problem 20ES ### 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 (the set of all subsets of X). A relation E is defined on P ( X ) as follows: For every A , B ∈ P ( X ) .

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 (the set of all subsets of X ). A relation E Is defined on P(X) as follows: For all A, B ∈ P(X), A E B ⇔ the number of elements in A equals the number of elements in B.

Calculation:

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

Let us consider that A=P(X)

The relation E Is given by

E={B,CP(X)|B has the same number of elements as C}

Reflexive:

The relation E Is reflexive if (a,a)E for every element aA.

Since A=P(X), E is reflexive if it contains (B,B) for all BP(X).

We note that E contains (B,B) because a set always has the same number of elements as itself and thus E is reflexive.

It is reflexive

Symmetric:

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

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