   # 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 ) = ( A × B ) ∪ ( A × C ) . ### 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 21ES
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 ) = ( A × B ) ∪ ( A × C ) .

To determine

Find the mistake in the following “proof”.

“Theorem” for all sets A and B,AcBc(AB)c.

“Proof:Suppose A and B are sets, and xAcBc by definition of union. It follows that xA and xB by definition of complement, and so xAB by definiton of union.

Thus x(AB)c by definiton of complement and hence AcBc(AB)c ”.

### Explanation of Solution

Given information:

Let A,B and C be any sets.

Concept used:

:Union of sets:Intersection of sets

: subset of set

Calculation:

Let A and B be two sets.

Prove that AcBc(AB)c.

Suppose there is an element x in AcBc, by definition of union, xAc or xBc.

By definition of complement it follows that xA or xB

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