Venn diagrams with two sets revision notes

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

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A two-set Venn diagram splits a group into the parts in each set, their overlap and neither set. The regions help you represent sets, read notation and find probabilities.

What union and intersection mean

Each person or item goes in exactly one region of the box. The intersection is the overlap, while the union is everything inside either circle. A ∩ B means in A and in B. It is only the overlap. A ∪ B means in A or B, or both. It is everything inside either circle. The intersection is the smallest region. The union is at least as large as either whole circle. The ∩ points down like the n in “and”. The ∪ is like a cup holding everything.

How to fill a diagram from totals

Start with the number in both sets, then subtract it from each set total to find the “only” regions. Finally, use the group total to find how many are in neither set. For 40 students, 25 study French, 18 study German and 7 study both. French only: 25 − 7 = 18. German only: 18 − 7 = 11. At least one language: 18 + 7 + 11 = 36. Neither: 40 − 36 = 4. n(A) means the whole of circle A, including both its regions.

How to read notation from the regions

A prime mark means “not in”. It describes a complement, which is everything outside a set. A ∪ B is everyone studying at least one language: 36 students. A ∩ B′ means in A but not B, so it is the French-only region: 18 students. For the pet example, (A ∪ B)′ means outside both circles. A′ ∩ B′ means not in A and not in B, so it describes the same region.

How to read a probability

For a randomly chosen person, divide the number in the required region by the total number in the box. The denominator includes people outside both circles. For 30 people, 12 own a dog, 9 own a cat and 5 own both. Dog only: 12 − 5 = 7. Cat only: 9 − 5 = 4. Neither: 30 − (7 + 5 + 4) = 14. At least one pet: 7 + 5 + 4 = 16. The probability of owning both is 5/30 = 1/6.

How to work backwards to find the overlap

If you know how many people are in neither set, find the union first. The two set totals count the overlap twice, so their sum exceeds the union by the overlap. For 50 people, 9 like neither, so 50 − 9 = 41 like tea or coffee. The set totals add to 32 + 21 = 53. Since 53 − 41 = 12, 12 people like both.