   # Use an element argument to prove each statement in 7-22. Assume that all sets are subsets of a universal set U. For all sets A , B , and C , ( A − B ) ∪ ( c − B ) = ( A ∪ C ) − B . ### 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 9ES
Textbook Problem
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## Use an element argument to prove each statement in 7-22. Assume that all sets are subsets of a universal set U. For all sets A,B, and C, ( A − B ) ∪ ( c − B ) = ( A ∪ C ) − B .

To determine

To prove:

For all sets A ,B and C, (A-B)(CB)=(AC)B

### Explanation of Solution

Given information:

A and B be two sets.

Concept used:

:Union of sets:Intersection of sets

Calculation:

Let A,B and C are any three sets.

Prove that (AB)(CB)=(AC)B

First prove that (AB)(CB)(AC)B.

Suppose there is an element x is (AB)(CB), by definition of union,

xAB or xCB

Case 1.

If xAB then by set difference xA and xB.

Since xA by definition of union it follows that xAC.

Thus, xAC and xB the definition of set difference implies that x(AC)B.

Hence, xABx(AC)B.

Case 2.

If xCB then by set difference xC and xB.

Since xC, by definition of union it follows that xAC.

Thus, xAC and xB, the definition of set difference implies that x(AC)B.

Hence xCBx(AC)B

Therefore, both the cases yields x(AC)B

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