SOLUTION: Prove that P(AuB)' = P(A'nB').

Question 1170006: Prove that P(AuB)' = P(A'nB').

Answer by ikleyn(52872) (Show Source): You can put this solution on YOUR website!
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Prove that P(AuB)' = P(A'nB').
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This equality,  P(AuB)' = P(A'nB'),  is true, because the equality for the subject sets

    (AuB)' = (A'nB')

is true.


Indeed, if element x of the universal set U belongs to (AuB)'  (left side set),  
then x does not belong to the union AuB, which means that x belongs NEITHER A NOR B.

Hence, x belongs to each set A' and B', which means that x belons to the intersection (A'nB')  
(right side set).



Inversely, if element x of the universal set U belongs to (A'nB')  (right side set),  
then x does belong to each set A' and B', which means that x belongs NEITHER A NOR B.

Hence, x belons to the complement of the union (AnB)'  (left side set).


Thus we proved that every element of the left side set belongs to the right side set,
and inversely, each element of the right side set does belong to the left side set.


Hence, the sets are equal, and, THEREFORE, the probabilities are equal, QED.

Solved.

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