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Date: 2-2-2016
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Date: 14-2-2017
1917
Date: 11-1-2022
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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) |
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(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.
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