   Chapter 9.3, Problem 46ES ### 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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# Prove the inclusion/exclusion rule for two sets A and B by showing that A ∪ B can be partitioned into A ∩ B , A − ( A ∩ B ) , and B − ( A ∩ B ) , and then the addition and difference rules. (See the hint for exercise 39 in Section 6.2.)

To determine

To prove the inclusion/exclusion rule for two sets A and B by showing that AB can be partitioned into AB,A(AB) and B(AB) and then using the addition and difference rules.

Explanation

Given information:

AB Denotes the set consists of all elements that belong to either set A or set B.

Concept used:

If the sets A,B are disjoint, then they do not have any common elements.

Therefore,

N(AB)=N(A)+N(B)

Calculation:

Let the sets A,B are not disjoint.

The set A(AB) consists of all elements that belong to A but not B.

The set B(AB) consists of all elements that belong to B but not A.

The set AB consists of all elements that belong to both A and B.

If an element x belongs to AB then only one of the following three cases is true:

1) The element x belongs to A but not B.

2) The element x belongs to B but not A.

3) The element x belongs to both A and B

The three sets A(AB),B(AB) and AB are pair wise disjoint sets.

Therefore,

AB=(A(AB))+(B(AB))(AB)

Therefore

N(AB)=N(A(AB))+N(B(AB))N(AB)..........(1)

Since the sets B(AB) & AB are disjoints and A(AB)&AB are disjoints, so

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