Explanation Given information: 1) For an arbitrary set U , the power set P ( U ) is defined by P ( U ) = { X | X ⊆ U } and addition in P ( U ) is defined by A + B = ( A ∪ B ) − ( A ∩ B ) A ⋅ B = A ∩ B Formula used: A ∪ B = { a | a ∈ A o r a ∈ B } A ∩ B = { a | a ∈ A a n d a ∈ B } Calculation: Let U = { a , b , c } be a set with 3 elements | P ( U ) | = 2 3 = 8 P ( U ) = { ∅ , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c } } . The addition and multiplication of a set is defined as A + B = ( A ∪ B ) − ( A ∩ B ) A ⋅ B = A ∩ B Therefore, Case (i) When both sets have the same elements, { a } + { a } = ( { a } ∪ { a } ) − ( { a } ∩ { a } ) = { a } − { a } = ∅ , { b } + { b } = ( { b } ∪ { b } ) − ( { b } ∩ { b } ) = { b } − { b } = ∅ and { c } + { c } = ( { c } ∪ { c } ) − ( { c } ∩ { c } ) = { c } − { c } = ∅ (ii) When one of the set is empty, { a } + ∅ = ( { a } ∪ { ∅ } ) − ( { a } ∩ { ∅ } ) = { a } − { ∅ } = { a } ; { b } + ∅ = ( { b } ∪ { ∅ } ) − ( { b } ∩ { ∅ } ) = { b } − { ∅ } = { b } ; { c } + ∅ = ( { c } ∪ { ∅ } ) − ( { c } ∩ { ∅ } ) = { c } − { ∅ } = { c } In similar way, addition of one empty and other non-empty set is the same as the non-empty set itself. (iii) When both sets are disjoint, { a } + { b } = ( { a } ∪ { b } ) − ( { a } ∩ { b } ) = { a , b } − { ∅ } = { a , b } ; { a } + { c } = ( { a } ∪ { c } ) − ( { a } ∩ { c } ) = { a , c } − { ∅ } = { a , c } ; { b } + { c } = ( { b } ∪ { c } ) − ( { b } ∩ { c } ) = { b , c } − { ∅ } = { b , c } . In similar way, addition of two disjoint sets is the union of those two sets. (iv) If one set is a proper subset of other, Let { a } ⊂ U , then { a } + U = ( { a } ∪ U ) − ( { a } ∩ U ) = U − { a } = { b , c } . If { b } ⊂ U , then { b } + U = ( { b } ∪ U ) − ( { b } ∩ U ) = U − { b } = { a , c } . If { c } ⊂ U , then { c } + U = ( { c } ∪ U ) − ( { c } ∩ U ) = U − { c } = { a , b } . That is, if one set is a proper subset of other, then their addition is the same as the subtraction of elements of subset from the elements of superset. (iv) Two different sets with non-empty intersection, Let { a , b } , { a , c } be such two sets; then ( { a , b } ∪ { a , c } ) − ( { a , b } ∩ { a , c } ) = { a , b , c } − { a } = { b , c } Therefore, 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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Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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Chapter 5.1, Problem 4E

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To construct: Addition and multiplication table for the ring P(U), U={a,b,c}.

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Chapter 5.1, Problem 3E

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Let S = {4, 7, 41, 74}. Recall that the power set P (S) of S is the set of all subsets of the set S. Notice that (P (S);A,n) is a commutative ring with unity, where AAB = (A – B)U (B – A) for any A, BE P (S) and -, U,n are the usual set difference, union, and intersection, respectively. List the elements of the following sets: a. The additive identity, which is the element with respect to A, is the set b. The the additive inverse of A = {4}, which is the inverse of A with respect to A, is the set c. The unity is the set d. The set S = {4,7, 41, 74} is ? v since the additive identity the unity e. The set A = {4} is ? v since An ? f. The ring (P (S);A,n) is ?

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Elements Of Modern Algebra

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To construct: Addition and multiplication table for the ring P ( U ) , U = { a , b , c } .

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To construct: Addition and multiplication table for the ring P(U), U={a,b,c}.

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