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.`

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 **.

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 X be a non-empty set, then P(X)={A:AsubX} is a Boolean lattice w.r.t. the operations A+B=AcupB,A.B=A capB and A'= X-A for all A, B inP(X) The additive identity of this Boolean algebra is

Prove that P(A uu B) = P(A) + P(B) - P(A nn B)

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).

If A and B are two non-empty sets having n elements in common , then show that A xx B and B xx A have n^2 elements in common.

If a^(x) = b^(y) = c^(z) and a, b, c, are in G.P., prove that x, y, z are in H.P.

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.

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).