Given a non -empty set X, let `*:" "P(X)" "xx" "P(X) ->P(X)` be defined as A * B = (A B) `uu` (B A), `AA` A, B ` in ` P(X)`A" "*" "B" "=" "(A" "-" "B) uu(B" "-" "A), AA""""A ," "B in P(X)` . Show that the empty set `varphi` is the identity for the

Given a non - empty set, X , consider the binary operation ** : P(X) xxP(X) rarr P(X) given by A **B = Acap B , AAA,B in P(X) , where P(X) is the power set 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 4P(A) = 6 P(B) = 10 P(A cap B) = 1 then P(B | A) = ……….

For events A and B if P(A) = 0.3, P(B) = 0.6 and P(B | A) = 0.5 then find P( A | B) and P(A cup B) .

If P(A') = 0.7, P(B) = 0.7, P(B | A) = 0.5 then find P ( A | B) and P(A cup B) .

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

If P(A) = 0.4, P(B) = 0.8, P( B | A) = 0.6 then find P( A | B) and P(A cup B) .

Given two independent events A and B such that P(A)= 0.3, P(B)= 0.6 . Find P(A or B)

If P(A) = 0.5, P(B) = 0.6 and P(A cup B) = 0.8 then find P( A | B) and P( B | A) .