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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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Write out the elements of P ( A ) for the set A = { a , b , c } , and construct an addition table for P ( A ) using addition as defined in Exercise 42 . (Sec. 1.1 , # 7 c )

Sec. 1.1 , # 7 c

42. For an arbitrary set A , the power set P ( A ) was defined in Section 1.1 by P ( A ) = { X | X A } , and addition in P ( A ) was defined by

X + Y = ( X Y ) ( X Y )

= ( X Y ) ( Y X )

Prove that P ( A ) is a group with respect to this operation of addition.

If A has n distinct elements, state the order of P ( A ) .

To determine

To construct: An addition table of P(A) for the set A={a,b,c}.

Explanation

Given information:

For an arbitrary set A, the power set P(A) defined by P(A)={X|XA} and addition in P(A) defined by

X+Y=(XY)(XY)=(XY)(YX)

Explanation:

Let, A={a,b,c} set of 3 elements, |P(A)|=23=8.

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

The addition of set is defined as

X+Y=(XY)(XY)=(XY)(YX)

This addition can be performed through various cases.

Case 1. When both sets have the same elements:

Let X={a},Y={a}; then

X+Y=({a}{a})({a}{a})={a}{a}={}

Similarly, when both X and Y have the same element, their addition is an empty set.

Case 2. When one of the sets is empty:

Let X={a},Y={}; then

X+Y=({a}{})({a}{})={a}{}={a}

Similarly, addition of one empty and other non-empty set is the same as the non-empty set itself.

Case 3. When both sets are disjoint:

Let X={a},Y={b}; then

X+Y=({a}{b})({a}{b})={a,b}{}={a,b}

Similarly, addition of two disjoint sets is the union of those two sets.

Case 4. If one set is a proper subset of other:

Let X={a},Y={a,b}, then

X+Y=({a}{a,b})({a}{a,b})={a,b}{a}={b}

That is, when one set is a proper subset of other, then their addition is the same as the subtraction of elements of the subset from the elements of the superset.

Case 5. Two different sets with non-empty intersection:

Let X={a,b},Y={a,c}; then

X+Y=({a,b}{a,c})({a,b}{a,c})={a,b,c}{a}={b,c}

When two sets are different with non-empty intersection, then their addition is all elements of both sets, except the common elements

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