   # Prove each statement is 39-44. For all sets A and B . ( A − B ) ( B − A ) ∪ ( A ∩ B ) = A ∪ B The sets ( A − B ) , ( B − A ) , and ( A ∩ B ) are mutually disjouint. ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
Chapter 6.2, Problem 39ES
Textbook Problem
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## Prove each statement is 39-44.For all sets A and B. ( A − B ) ( B − A ) ∪ ( A ∩ B ) = A ∪ B The sets ( A − B ) , ( B − A ) , and ( A ∩ B ) are mutually disjouint.

To determine

(a)

To Prove:

(AB)(BA)(AB)=AB

### Explanation of Solution

Given information:

Let A and B be any sets.

Concept used:

:Union of sets:Intersection of sets

: subset of set

: Null set

Calculation:

Let A and B be two Sets.

Prove that (AB)(BA)(AB)=AB.

First we prove that (AB)(BA)(AB)AB.

Observe the following conditions:

If xAB that implies xA and xB.

If xBA that implies xB and xA

If xAB that impliesxA and xB

The set AB contains all the elements or the set A and of the set B.

So ¡f an element x satisfies any one of the above three conditions then x must be in the union of A and B.

Therefore, (AB)(BA)(AB)AB

To determine

(b)

To Prove:

The sets (AB),(BA) and (AB) are mutually disjoint.

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