   Chapter 1.4, Problem 3TFE

Chapter
Section
Textbook Problem
1 views

# Label each of the following statements as either true or false.Let Α = { a ,   b ,   c } . The power set P ( A ) is closed with respect to the binary operation ∩ of forming intersections.

To determine

Whether the statement, “Let A={a,b,c}. The power set P(A) is closed with respect to the binary operation of forming intersections” is true or false.

Explanation

Formula used:

Suppose that * is a binary operation on a non-empty set A, and let BA. If x*y is an element of B for all xB and yB, then B is closed with respect to *.

Explanation:

Let A={a,b,c}

By using the definition of power set,

P(A)={,{a},{b},{c},{a,b},{a,c},{b,c},A}

(i) When both sets have the same elements,

{a}{a}=({a}{a})={a} ;

{b}{b}=({b}{b})={b} ;

{c}{c}=({c}{c})={c}

In a similar way, when both sets have the same element, their intersection is the set itself.

(iii) When one of the sets is empty.

{a}={a})= ;

{b}={b})= ;

{c}={c})=

Similarly, intersection of one empty and other non-empty set is the same – always an empty set.

(iii) When both sets are disjoint.

{a}{b}=({a}{b})= ;

{a}{c}=({a}{c})= ;

{b}{c}=({b}{c})=.

In a similar way, intersection of two disjoint sets is the empty set.

(iv) If one set is a proper subset of other.

{a}{a,b}=({a}{a,b})={a} ;

{b}{b,c}=({b}{b,c})={b} ;

{c}{a,b,c}=({c}{a,b,c})={c}.

That is, if one set is a proper subset of other, then their intersection is the same as the proper subset.

(v) Two different sets with non-empty intersection.

{a,b}{a,c}=({a,b}{a,c})={a}{a,b}{b,c}=({a,b}{b,c})={b};{a,c}{b,c}=({a,c}{b,c})={c};

Thus, when two sets are different with non-empty intersection, then their intersection is all common elements of both the sets

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### In Exercises 99106, factor out the greatest common factor from each expression. 102. 3x2/3 2x1/3

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### True or False: If f′(c) = 0 and f″(c) < 0, then c is a local minimum.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

#### What are the five steps of the scientific method?

Research Methods for the Behavioral Sciences (MindTap Course List)

#### How does a full-text database differ from other databases?

Research Methods for the Behavioral Sciences (MindTap Course List) 