For a given set S, let P(S) be the collection of all subsets of S. Let binaryoperations + and on P(S) be defined byA+ B = (AU B) – (An B) and A - B = An Bfor A, B E P(S).(a) Give the tables for + and for P({x, y}).(b) Prove that for any set S, (P(S), +,) is a ring.(e) Prove or disprove that P({z, y} , +,·) is a commutative ring withunity.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 23E: 23. Let be the equivalence relation on defined by if and only if there exists an element in ...

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Label each of the following statements as either true or false. The least upper bound of a nonempty set S is unique.
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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. 1.1,7c) Sec. 1.1,7c 42. For an arbitrary set A, the power set P(A) was defined in Section 1.1 by P(A)={ XXA }, and addition in P(A) was defined by X+Y=(XY)(XY) =(XY)(YX) 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).
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