## 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.