   # 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 22ES
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” that for all sets A and B, (AB)(AB)A.

Proof; Suppose A and B are sets, and suppose x(AB)(AB). If xA then xAB.

Then, by definition of difference, xA and xB. Hence xA, and so (AB)(AB)A by definition of subset.

### Explanation of Solution

Given information:

Let  A and B be any set

Concept used:

:Union of sets:Intersection of sets

: subset of set

Calculation:

Consider the statement

Proof suppose A and B are sets, and suppose x(AB)(AB). If xA then xAB. Then by definition of difference xA and xB. Hence xA, and so (AB)(AB)A by definition of subset.

The objective is to find the mistake in the above statement.

Suppose x(AB)(AB)

“If xA then xAB

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