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Question

Let X be a non- empty set and be a binary operation on P(x) defined by AB=AB for all A,BP(x). Find the identity and invertible elements

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Solution

Let X={1,2,3}
P(X)={ϕ,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}}
A x=AX=A
X A=XA=A
(i)e is the identity of
a X=AX=A for aA
X A=XA=A
A X=X A=A for AP(x)
Hence X is an identity element.
(ii)a b=e=b a
We have e=X
A B=e=B A
AB=X
This is possible if A=B=X
A X=A=X A for all AP(X)
Hence X is the only invertible element in P(X)

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