Problem:
Sets and , shown in the Venn diagram, have the same number of elements. Their union has elements and their intersection has elements. Find the number of elements in .
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Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let denote the set of elements that are in but not in . Let denote the set of elements that are in but not in . Because sets and have the same number of elements, the number of elements in is the same as the number of elements in . This number is half the number of elements in the union of and minus the intersection of and . That is, the number of elements in each of and is
Adding the number of elements in and to the number in but not in gives elements in .
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Let be the number of elements each in and . Then , and .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions