   Chapter 6.3, Problem 51ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Refer to the definition of symmetric difference given above. Prove each of 47-52, assuming that A, B, and C are all subsets of a universal set U. If A Δ C = B Δ C ,   then   A = B .

To determine

To prove:

If AΔC=BΔC,then A=B.

Explanation

Given information:

A,B and C be any sets.

Concept used:

AΔB=(AB)(BA).

Calculation:

The following are the Venn diagrams for AΔCandBΔC :

The shaded portions denote the area covered by the symmetric difference.

AΔC=(AC)(CA)BΔC=(BC)(CB)

From the Venn diagram, it can be seen that if an element xA

then it is present in either of the two regions:

xACxAC

Analyze both the above cases as follows:

Case 1: xAC

The following two things are observed in this case using the above Venn

diagram:

xC Because xAC

xAΔC Because the region where x belongs to is contained in the AΔC region.

Since it is given that AΔC=BΔC therefore xBΔC.

But xC, therefore, x must belong to B.

Case2: xAC

The following two things are observed in this case using the above Venn diagram:

xCbecausexAC

xAΔCbecause the region where xbelongs to is not contained in the AΔCregion.Since, it is given that AΔC=BΔC,therefore x(BΔC)either.

But xC, therefore, x must belong to BC because that is the only region that contains C

But is itself not contained in (BΔC) therefore xB.

So, it is concluded that every element that belongs to A also belongs to B.

Thus, A is a subset of B

AB

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