   Chapter 8.2, Problem 23ES ### 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 answers. Let X be a nonempty set and φ ( X ) the power set of X. Define the “subset” relation S on φ ( X ) as follows: For every A , B ∈ φ ( X ) ,   A   S   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 “subset” relation S on P(X) as follows: For all A, B ∈ P(X), A S B ⇔ A ⊆ B.

Calculation:

X= nonempty set

A=P(X)

S={B,CP(X)|BC}

Reflexive:

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

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

We note that S contains (B,B) because a set is always a subset of itself B and thus S is reflexive.

It is reflexive

Symmetric:

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

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