Sets and Venn Notation
A set is a collection of elements, and set notation is a compact way of describing which elements belong where. The universal set, written with the symbol for xi, contains everything under discussion; the intersection of A and B contains the elements in BOTH sets; the union contains the elements in EITHER set or both; and the complement of A, written A with a dash, contains everything in the universal set that is not in A. The notation n(A) means the number of elements in A. Venn diagrams make these relationships visible, and the counting rule n(A union B) = n(A) + n(B) - n(A intersection B) follows directly, because the overlap would otherwise be counted twice.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Learn the symbols precisely: the cup shape is union (either or both), the cap shape is intersection (both), a dash after a set is its complement (not in that set), the epsilon-like symbol means is an element of, and n(A) is the number of elements in A.
- When filling a Venn diagram, always start with the intersection, the middle region, and work outwards. Every other region is then found by subtraction, which prevents double counting.
- For two sets, the four regions are: in both, in A only, in B only, and in neither. Check that the four region totals add up to the number in the universal set.
- Translate the notation into a region before shading: A intersection B dash means in A but not in B, and (A union B) dash means in neither.
- Use the counting rule n(A union B) = n(A) + n(B) - n(A intersection B) when the question gives totals rather than a diagram, and rearrange it when the overlap is the unknown.
- For a probability question built on a Venn diagram, count the elements in the region asked for and divide by the number in the universal set, and read carefully whether the question restricts attention to a smaller group.
Worked example
In a class of 30 students, 18 study French, 15 study Spanish and 7 study both. Complete a Venn diagram and find how many study neither language.
- Start with the intersection: 7 students study both, so 7 goes in the overlap.
- French only = 18 - 7 = 11, because the 18 who study French includes the 7 who study both.
- Spanish only = 15 - 7 = 8, by the same reasoning.
- Add the three filled regions: 11 + 7 + 8 = 26 students study at least one language.
- Subtract from the universal set: 30 - 26 = 4 study neither, and that number goes outside both circles but inside the rectangle.
- Check with the counting rule: n(F union S) = 18 + 15 - 7 = 26, which matches, so the diagram is consistent.
Practice questions
Try each question, then tap to reveal the answer.
Q1What does n(A) mean?Show answer
Answer: The number of elements in set A.
Q2Describe in words the region A intersection B.Show answer
Answer: The elements that are in both A and B.
Q3n(A) = 12, n(B) = 9 and n(A intersection B) = 4. Find n(A union B).Show answer
Answer: 12 + 9 - 4 = 17.
Q4In a universal set of 40, n(A union B) = 31. How many elements are in neither A nor B?Show answer
Answer: 40 - 31 = 9.
Q5Describe the region shaded when A dash intersection B is shaded.Show answer
Answer: The elements that are in B but not in A.
Q6Why is the intersection filled in first when completing a Venn diagram?Show answer
Answer: Because the totals given for each set include the overlap, so every other region is found by subtracting the overlap. Starting elsewhere leads to double counting.
Q7A student picks one of the 30 class members at random. Using the worked example above, what is the probability they study French only?Show answer
Answer: 11 out of 30, which is 11/30.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
In a survey of 50 people, 28 own a dog, 22 own a cat and 9 own both. (a) Draw a Venn diagram to show this information. (b) One person is chosen at random. Find the probability that they own exactly one of the two animals.
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Sets A and B are such that n(A) = 20, n(B) = 14 and n(A union B) = 27. (a) Find n(A intersection B). (b) Hence find the number of elements in A but not in B.
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Free printable worksheet
Want more practice on paper? Download the sets and venn notation worksheet pack - 10 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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