Let X be a non-empty set and P(X) be a power set Let * be a binary operation defined - Maths -

Question

Let X be a non-empty set and P(X) be a power set Let * be a binary
operation defined on the elements of P(X) by A*B = A intersection
B, for all A,B belonging to P(X)
(i) Check whether * is a binary operation
(ii) Verify whether * is associative on P(X)
(iii) Find the identity elements of P(X)
(iv) Find all invertible elements of P(X)

Lovina Kansal · Accepted Answer

Dear student
iii) Let E be the identity element in P(X) with respect to *
Then,A*E=A=E*A for all A∈P(X)A∩E=A=E∩A for all A⊂XE=XThus, X is the identity element with respect to * on P(X)
iv)Let A be an invertible element of P(X) and let S be its inverse
Then,A*S=X=S*AA∩S=X=S∩AA=S=XThus, X is the only invertible element of P(X) with respet to * and it is the inverse ofitself
 
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