11. Show that if A and B are sets, then (a) A − B = A ∩ B (b) (A ∩ B) ∪ (A ∩ B) = A
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a)
We must show and
Show that
Let By definition of set difference, and By definition of complement, implies that Hence, it is true that both, and By definition of intersection,
Show that
Let By definition of intersection, and By definition of complement, implies that Hence, and By definition of set difference,
Thus,
b)
Let There are two cases, and or and
In the first case so by definition set difference,
In the second case and so by definition of intersection
By definition of union
Thus if then
Let This means that either or
In the first case in the second case and
Then in either case
Two sets are equal, since they have the same elements.Therefore