sets - help reqd..?

Question

if X & Y are independent, show that X and Y-complement are independent.

Answer 1

x complement are those elements in the universal which are not in x and y complement are those elements in the universal which are not in y
so x' and y' are disjoint
let U={1,2,3,4,5,6,7,8,9,10}
letx={1,2,3,4}
y={7,8,9}
x'={5,6,7,8,9,10}
y'={1,2,3,4,5,6,10}
x' and y' are disjoint

Answer 2

Since X and Y are independent, P(X and Y) = P(X)P(Y).

We now need to show that P(X and ~Y) = P(X)P(~Y) (where ~Y refers to the complement of Y)

P(X)P(~Y) = P(X)(1-P(Y)) = P(X) - P(X)P(Y) = P(X) - P(X and Y),
because X and Y are independent.

By the inclusion-exclusion principle,
P(X) = P((X and Y) or (X and ~Y))
= P(X and Y) + P(X and ~Y) - P((X and Y) and (X and ~Y)),
but Y and ~Y cannot simultaneously be true, so
P((X and Y) and (X and ~Y)) = 0.

Therefore
P(X) = P(X and Y) + P(X and ~Y), meaning
P(X) - P(X and Y) = P(X and ~Y).

But P(X)P(~Y) = P(X) - P(X and Y), from above, so
P(X)P(~Y) = P(X and ~Y), meaning X and ~Y are independent.