If n{A} = 6, n{B} = 5 and n{A ∩ B} = 2, find n{A ∪ B}

Analyze the question: You are given the sizes of two sets and the size of their overlap, and you must find the size of their union.

Concept: When you add n(A) and n(B), the elements in the intersection get counted twice. So you subtract the intersection once:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Solution:

• n(A) = 6
• n(B) = 5
• n(A ∩ B) = 2

n(A ∪ B) = 6 + 5 − 2 n(A ∪ B) = 11 − 2 n(A ∪ B) = 9

Check: Only A has 6 − 2 = 4 elements, only B has 5 − 2 = 3 elements, and both have 2. Total = 4 + 3 + 2 = 9. ✔

Tip: Remember "add both, subtract the overlap." Whenever you see union and intersection in the same question, think of the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Drawing a quick Venn diagram and filling in the overlap first also helps you avoid mistakes. Option D (13) is the trap if you forget to subtract, and 11 is what you get by only adding.