1. Let X be a set and P(X) be its power set. P (X) is a commutative ring with unity with the following operations + and

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1 Let X Be A Set And P X Be Its Power Set P X Is A Commutative Ring With Unity With The Following Operations And 1 (56.92 KiB) Viewed 25 times

1. Let X be a set and P(X) be its power set. P (X) is a commutative ring with unity with the following operations + and defined by: A+B : = (AUB) \(ANB) A.B : ANB for A, B E P(X). (Show it). Find Char(P (X)). Hint: The zero element of P (X) is Ø, since A+= (AU) \(Anø= A=0+ A. The unity of P(X) is X, since An X = A= AnX, for all A E P(X). Char(P (X)) = 2, since 2A = A+A= ( AA) U (AA) = 0. Also (P(X), +..) is a Boolean ring, since every element of P(X) is idempotent; that is An A= A, for all A EP (X). Therefore, Char(P (X)) = 2. Remark: If Char(R) = 0, then the ring has infinitely many elements. But the converse is not true. For example, the ring P(Z) which has infinitely many elements, but the Char(P (Z)) = 2. =