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For each set S, let peS) denote the set of all subsets of S. For A ,B E P(S), define A +

Chapter 24, Problem 24.22

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QUESTION:

For each set S, let peS) denote the set of all subsets of S. For A ,B E P(S), define A + B andABbyA + B 0: (A U B)\(A n B) = {x: x E AU B and x fj A n B)andAB=AnB.Verify that with these operations, F'(S) is a ring. [peS) is called the power set of A. SeeAppendix A for facts about sets. Suggestion: Make generous use of Venn diagrams in thisproblem.)

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QUESTION:

For each set S, let peS) denote the set of all subsets of S. For A ,B E P(S), define A + B andABbyA + B 0: (A U B)\(A n B) = {x: x E AU B and x fj A n B)andAB=AnB.Verify that with these operations, F'(S) is a ring. [peS) is called the power set of A. SeeAppendix A for facts about sets. Suggestion: Make generous use of Venn diagrams in thisproblem.)

ANSWER:

Step 1 of 8

Let  be a non-empty set and  be the set all subsets of .

Show that  is a ring with the following operations

It is enough to show that  is Abelian group and  semi-group and also satisfies the distributive properties.

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