Given a non-empty set X, consider the binary opertion `**: P(X) xx P (Y) to P (X)` given by `A **B= A nnB AA A, B` in `P (X),` where P(X) is the power set of X. Show that X is the identity element for this opertion and X is the only invertible elemtn in P(X) with respect to the opertion `**.`

Given a non-empty set X, let **: P(X) xx P (X) to P (X) be defined as A **B = (A-B) uu (B-A) , AA A, B in P (X). Show that the empty set phi is the identity for the opertion ** and all the elements A of P (X) are invertible with A ^(-1) =A.

Show that -a is the inverse of a for the addition opertion '+' on R and 1/a is the inverse of a ne 0 for the multiplication opertion 'x' on R.

For any two sets A and B, prove that P(A cap B) = P(A) capP(B) where P(A) is the power set of A.

Let S be a non-empty set and P(S) be the power set of the Set S. Statement -I: Phi is the identity element for union as a binary operation on P(S) Statement -II: S is the identity element for intersection on P(S).

For any two sets A and B, if P(A)=P(B), then show that A=B, here P(A) is the power set of A.

Let P(A) b the power set of non-empty set A. A binary operation ** is defined on P(A) as X**Y=XcapY for all X,YinP(A) . Detemine the identity element In P(A). Prove that A is the only invertible element in P(A).

Let P(C ) be the power set of non-empty set C. A binary operation ** is defined on P(C ) as A**B=(A-B)cup(B-A) for all A,BinP(C ) . Prove that Phi is the identity element for ** on P(C ) and A is invertible for all AinQQ_(0) .

Define a binary opertion ** on the set {0,1,2,3,4,5} as a**b ={{:(a+b"," , if a + b lt 6), ( a + b -6, if a + b ge 6):} Show that zero is the identity for this opertion and each element ane 0 of the set is invertible with 6-a being the inverse of a.

Let P(A) be the power set of a non-empty set A and a binary operation @ on P(A) is defined by X@Y=XcupY for all YinP(A). Prove that, the binary operation @ is commutative as well as associative on P(A). Find the identity element w.r.t. binary operation @ on P(A). Also prove that Phi inP(A) is the only invertible element in P(A).

Given a non empty set X, consider P (X) which is the set of all subsets of X. Define the relation R in P(X) as follows : For subsets A, B in P(X), ARB if and only if A sub B. Is R an equivalence relation on P (X)? Justify your answer.