   # Find the mistake in the following “proof” that for all sets A and B. ( A − B ) ∪ ( A ∩ B ) ⊆ A . “Proof: Suppose A and B are any sets, and suppose x ∈ ( A − B ) ∪ ( A ∩ B ) . If x ∈ A then x ∈ A − B , and so, by definition of difference, x ∈ A and x ∈ B . In particular, x ∈ A , and, therefore, ( A − B ) ∪ ( A ∩ B ) ⊆ A by definition of subset.” ### 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 25ES
Textbook Problem
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## Find the mistake in the following “proof” that for all sets A and B. ( A − B ) ∪ ( A ∩ B ) ⊆ A . “Proof: Suppose A and B are any sets, and suppose x ∈ ( A − B ) ∪ ( A ∩ B ) .If x ∈ A then x ∈ A − B , and so, by definition of difference, x ∈ A and x ∈ B . In particular, x ∈ A , and, therefore, ( A − B ) ∪ ( A ∩ B ) ⊆ A by definition of subset.”

To determine

To Prove:

For all sets AandB,(AB)(ABc)=ϕ ..

### Explanation of Solution

Given information:

Let  A and B be any set

Concept used:

:Union of sets:Intersection of sets

: subset of set

Calculation:

Suppose not that is, suppose there exist sets AandBsuch that(AB)(ABc)=ϕ

Then there is an element x in (AB)(ABc) by definition of intersection,

xABandxABc

Applying the definition of intersection again we have tha

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