If y=-x2+14x+ then x=10 is a counter example for which conjecture
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a. The events are mutually exclusive, so
b. For the same reason,
Another way to see this is to use the result from part (a), and the inclusion/exclusion principle:
and the right side reduces to 0.
c. Same as (b),
d. Same as (b),
We can also use the distributivity rule for unions and intersections to write
e. If the is the complement of
, then by DeMorgan's law,