Counting elements in sets revision notes

Edexcel International GCSE Maths (4MA1), chapter 18: Sets, specification point 1.5c. Free lesson.

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The notation n(A) tells you how many elements are in a set, while Venn diagram rules help you count unions and complements without missing or double-counting elements.

What n(A) counts

The notation n(A) means the number of elements in set A. It gives a count, not a list of the members. It counts the whole of circle A, including any part in the overlap. n(A) is the total inside circle A, not just the part belonging only to A. n(A ∩ B) counts the overlap, the elements in both sets. n(ξ) counts everything in the box.

How to count regions in a Venn diagram

For 30 students, 18 play football, 14 play tennis and 8 play both. Put the overlap in first. Then subtract it from each circle's total to find the parts belonging only to one set. The remaining students are outside both circles. The whole of circle A is football only plus the overlap, so n(A) = 18. The union A ∪ B is everyone in either circle, counted once. Check that all the regions add to the total in the box.

How to count a union

Adding n(A) and n(B) counts the overlap twice, because those elements belong to both sets. Subtract the overlap once so everyone in the union is counted exactly once. n(A ∪ B) = n(A) + n(B) − n(A ∩ B). For the football and tennis example, 18 + 14 would give 32, which is more than the 30 students in the box. Subtracting the 8 who play both gives 24.

How to count a complement

A complement means everything in the box that is not in the named set. A dash after a set, such as A′, means its complement. You need the total in the box to count it. n(A′) = n(ξ) − n(A). (A ∪ B)′ means not in either circle, so it counts the region outside both.

Check the counts against the box

After finding the regions, add them together. They should equal n(ξ), the total number of elements in the box. This helps you spot a region that has been counted twice or placed incorrectly. For n(ξ) = 40, n(A) = 22, n(B) = 15 and n(A ∩ B) = 9: n(A ∪ B) = 22 + 15 − 9 = 28. n((A ∪ B)′) = 40 − 28 = 12.