Suppose Set B contains 45 elements and the total number elements in either Set A or Set B is 48. If the Sets A and B have 7 elements in common, how many elements are contained in set A?

Answer 1

B u A = 48

B = 45

The part of set B not included in A is B - A∩B = 45 - 7 = 38

Since B u A = 48, A must have 48-38 = 10 elements (3 exclusive of B, 7 shared with B).

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Alternatively, the inclusion-exclusion principle may be used: A + B - (A ∩ B) = A u B

A + 45 - 7 = 48

A = 48 - 45 + 7

A = 10

NOTE that I'm using A and B to indicate the number of elements in each set, respectively.

More formally, one would write |A| and |B| to indicate the cardinality (number of elements) of the set.

Answer 2

the Venn Diagram really helps.

B has 45 elements and 7 of them are also in A.

So 38 of them are exclusive to B.

38 + 7 + a = 48

45 + a = 48

a = 3

Alternatively, you can simply argue that since there are 48 of them altogether and

B has 45, that leaves 3 EXCLUSIVELY for set A.

So A has 3+7 = 10 elements