   Chapter 6.3, Problem 45ES ### 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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# Consider the following set property: For all sets A, B, and C, ( A − B ) ∪ ( B − C ) = ( A ∪ B ) − ( B ∩ C ) . Use an element argument to derive the property. Use an algebraic argument to derive the property (by applying properties from Thorem 6.2.2). Comment on which method you found easier.

To determine

(a)

To derive:

Use an element argument to derive the property.

Explanation

Given information

(AB)C=(AC)(BC).

Concept used:

(AB)C=(AC)(BC).

Cite a property from Theorem 6.2.2 for every step of the proof.

Solution: Let A,B,and C be any sets. Then

(AB)C=(AB)Ccby the set difference law=Cc(AB)by the commutative law for =(CcA)(CcB)by the distributive law=(ACc)(BCc)by the commutative law for =(AC)(BC)by the set difference law.

Calculation:

Consider the set property,

(AB)(BC)=(AB)(BC), for any three sets A,B,andC.

(a)

Derive the stated property: (AB)(BC)=(AB)(BC) in the method of element

argument.

In element argument method, a property P=Q can be shown, by showing

PQ,and QP.

To show (AB)(BC)(AB)(BC) :

Let x(AB)(BC). Then, by the definition of set union, x(AB)or(BC).

Case - (i) :

When x(AB), by the definition of set difference, xAandxB.

Since xA and by the definition of set union, xAB. Also, since xB and by the definition

Of set interaction, xBC.

Finally, since

xAB,and xBC,and by the definition of set difference,x(AB)(BC)

To determine

(b)

To derive:

Use an algebraic argument to derive the property (by applying properties from Theorem).

To determine

(c)

To derive:

Comment on which method you found easier.

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