   Chapter 6.3, Problem 13ES ### 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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# For each of 5—21 prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set ∪ . 13. For all sets A, B, and C, A ∪ ( B − C ) = ( A ∪ B ) − ( A ∪ C ) .

To determine

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set .

For all sets A,B and C ,

A(BC)=(AB)(AC)

Explanation

Given information:

Let  A,B and C  be any set

Concept used:

:Union of sets:Intersection of sets

: subset of set

Calculation:

Suppose X and Y are two subsets of a universal set U. From the definition of differences and unions,

• The difference Y minus X is denoted by YX and is defined as the set of all elements that are present in Y but not in X.
• Also, the union X and Y is denoted by XY and is defined as the set of elements that are present in at least either X or Y.

Consider the statement,

A(BC)=(AB)(AC)

The above statement is false.

Consider a counter example where A={1,2,3},B={3,4,5} and C={2,4,6}.

Evaluate A(BC) corresponding to the counter example.

By the definition of difference.

BC={3,4,5}{2,4,6}={3,5}

Next find A(BC).

By the definition of union.

A(BC)={1,2,3}{3,5}={1,2,3,5}

Evaluate (AB)(AC) corresponding to the counter example

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