Set A consists of the integers from 4 to 12, inclusive, while set B consists of the integers from 6 to 20, inclusive. How many distinct integers do belong to the both sets at the same time?
A. 5
B. 7
C. 8
D. 9
E. 10

(B) Set A and set B are overlapping. To determine the number overlapping elements determine the points where the overlapping begins and ends: The sets begin to overlap at 6 and end at 12. There are 12 – (6 – 1) = 7 integers between 6 and 12 inclusive. The correct answer is B.
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I understand that set A and set B are composed of 5, 6, 7, 8, 9, 10, 11 and 7, 8, 9, 10, 11 ... so on, respectively. So they are supposed to have 5 integers in common (7, 8, 9, 10, 11), aren't they? What do I miss?

You've missed the word "inclusive" in the question statement, which means that the border values are also the elements of the sets.

Therefore the sets are:
{4, 5, 6, 7, 8, 9, 10, 11, 12}
{6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}\

The intersection of the sets is
{6, 7, 8, 9, 10, 11, 12}
It contains 7 elements.