   Chapter 8.2, Problem 24ES ### 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 be a nonempty set and P ( X ) the power set of X. Define the “not equal to” relation U on P ( X ) as follows: For every A , B ∈ P ( X ) , A ∪ B ⇔ A ≠ B .

To determine

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

Explanation

Given information:

Let X be a nonempty set and P(X) the power set of X. Define the “not equal to” relation U on P(X) as follows: For all A, B ∈ P(X), A U B ⇔ AB.

Calculation:

X= nonempty set

A=P(X)

U={B,CP(X)BC}

Reflexive:

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

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

We note that U does not contain (B,B) because a set is never unequal to itself (B=B) and thus U is not reflexive.

Symmetric:

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

Let us assume that (B,C)U

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