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 operation **: P(X) xx P (X) 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 operation and X is the only invertible element in P(X) with respect to the operation **.

Assume that P ( A ) = P ( B ) . Show that A = B

If A = phi the empty set, then write the number of elements in P(A).

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 operation and each element ane 0 of the set is invertible with 6-a being the inverse of 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.

If 0 lt P(A) lt 1, 0 lt P(B) lt 1 and P(A cup B)=P(A)+P(B)-P(A)P(B) , then

If P(A) = P(B) = x and P(A cap B) = P(A^(') cap B^(')) = 1/3 then x =

For any sets A and B, show that P ( A ∩ B ) = P ( A ) ∩ P ( B ) .