   # Find the mistake in the following : proof” that for all sets A , B , and C , if A ⊆ B , there is an element x such that x ∈ A and x ∈ B , and since B ⊆ C , there is an element x such that x ∈ B and x ∈ C . Hence there is an element is an element x such that x ∈ A and x ∈ C . Hence there is an element x such x ∈ A and x ∈ C and 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 23ES
Textbook Problem
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## Find the mistake in the following : proof” that for all sets A,B, and C, if A ⊆ B , there is an element x such that x ∈ A and x ∈ B , and since B ⊆ C , there is an element x such that x ∈ B and x ∈ C . Hence there is an element is an element x such that x ∈ A and x ∈ C . Hence there is an element x such x ∈ A and x ∈ C and A ⊆ C " .

To determine

Fill in the blanks in the following proof that for all sets A and B, (AB)(BA)=0.

Proof: Let A and B be any sets and suppose (AB)(BA)=0. That is, suppose there were an element x in (a). By definition of (b), xABandx(c). Then by definition of set difference, xA and xB and x(d) and x(e). In particular xA and x(f), which is a contradiction. Hence [the supposition that (AB)(BA)=0 is false, and so] (g).

### Explanation of Solution

Given information:

Let  A and B be any set

Concept used:

:Union of sets:Intersection of sets

: subset of set

Calculation:

Let A and B be any sets.

The objective is to fill the given blanks using the proof of the result (AB)(BA)=0 for all sets A and B.

Prove the result using contradiction method.

Suppose that (AB)(BA)=.

Let x(AB)(BA)

By the definition of intersection, x(AB) and x(BA)

Then by the definition of set difference, (xAandxB) and (xBandxA)

In particular, xAandxA, which is a contradiction.

Thus, the supposition that (AB)(BA)= is wrong.

Hence, the result is (AB)(BA)= for all A and B.

Use above proof for the result (AB)(BA)=, to fill the given blanks.

Fill the blanks as below,

Let A and B be any sets and suppose (AB)(BA)=

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