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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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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

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