Independence Complement Theorem

If sets and are independent, then so are and , where is the complement of (i.e., the set of all possible outcomes not contained in ). Let denote "or" and denote "and." Then

			(1)
			(2)

where is an abbreviation for . But and are independent, so

(3)

Also, since and are complements, they contain no common elements, which means that

(4)

for any . Plugging (4) and (3) into (2) then gives

(5)

Rearranging,

(6)

Q.E.D.