   Chapter 6.3, Problem 12ES ### 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 I true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.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 U.

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:

Consider the statement,

A(BC)=(AB)(AC)

For all sets A,B and C.

Here all sets are subsets of a universal set U.

The objective is to verify whether the statement is true or not.

Recall, the definition for Differences, and, Intersections.

Suppose A and B are two subsets of a universal set U.

1. The difference of B minus A is denoted by BA and is defined by, BA={xU|xB, and xA}
2. The intersection of A and B is denoted by AB, and is defined by

AB={xU|xA, and xB}

Let A,B and C be any sets

Consider the left hand side

A(BC)

Let xA(BC)

xA and x(BC) intersection of two setsxA and xB and xC Difference of two setsx(AB) and x(AC) Intersection of two setsx(AB)(AC) Difference of two sets

Thus,

A(BC)(AB)(AC)

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