X and Y are two sets such that n(X) = 15, n(Y) = 12 and n{X ∩ Y} = 7. Find ∩{X ∪ Y}

Analyze the question: You are given the sizes of two sets and the size of their intersection, and you need the size of their union. (The symbol "∩" in "∩{X ∪ Y}" is a typo for n, so the question is asking for n(X ∪ Y).)

Concept: The inclusion-exclusion principle for two sets:

n(X ∪ Y) = n(X) + n(Y) − n(X ∩ Y)

We subtract the intersection because the elements in both sets get counted twice when we add n(X) and n(Y).

Solution:

• n(X) = 15
• n(Y) = 12
• n(X ∩ Y) = 7

n(X ∪ Y) = 15 + 12 − 7 n(X ∪ Y) = 27 − 7 n(X ∪ Y) = 20

Answer: The union has 20 elements, which is option D.

Check: Only X has 15 − 7 = 8 elements, only Y has 12 − 7 = 5 elements, and both have 7. Then 8 + 5 + 7 = 20. ✓

Exam tip: Whenever you see a union, think "add, then subtract the overlap." If you forget to subtract, you get 27, which is a common trap. Drawing a quick Venn diagram and filling in the overlap first (7) makes it hard to go wrong.