   Chapter 7.1, Problem 50ES ### 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
1 views

# Given a set S and a subset A, the characteristic function of A, denoted X A , is the function defined from S to Z with the property that for each u ∈ S .

To determine

(a)

To prove:

For any two subsets A and B of S and each uS,

χAB(u)=χA(u)χB(u)

Explanation

Given information:

χA(u)={1ifuA0ifuA

Proof:

Consider that the characteristic function χA from Sto is defined as follows:

χA(u)={1ifuA0ifuA

If A and B are subsets of S then,

χAB(u)={1ifuAB0ifuAB

Now,

uAB

uAanduBχA(u)=1andχB(u)=1

χA(u)χB(u)=11=1

χAB(u)=1=11=χA(u)χB(u).............(1)

If uAB then either "uAanduB", "uAanduB" or "uAanduB"

To determine

(b)

To show:

For any two subsets A and B of S and each uS,

χAB(u)=χA(u)+χB(u)χA(u)χB(u).

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