Venn Diagrams and Set Notation
A Venn diagram uses overlapping circles inside a rectangle, the universal set, to show how different sets of items relate to each other: which items belong to one set, another, both, or neither, with each circle representing one named set and the overlap showing the intersection. Set notation gives this a shorthand, where n(A) means the number of elements in set A, the intersection of A and B means items in both sets, the union means items in at least one, and A' means every item not in A.
Before you start
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Method
- Draw a rectangle to represent the universal set (everyone or everything in the problem) and label it with the total number.
- Draw one circle inside the rectangle for each set described in the question, overlapping any circles that can happen together.
- Fill in the intersection (the overlapping region) first, since this is normally the number given directly, or the one found without needing to subtract anything else.
- Work outwards from the intersection: for each circle, subtract the intersection from that circle's total to find the 'only this set' region.
- Add up every region drawn inside the rectangle so far and subtract from the universal set total to find how many belong to neither set; this goes inside the rectangle but outside every circle.
- Check that every region you have written adds back up correctly to each given total and to the grand total.
- To find a probability from the diagram, count the elements in the region the question asks for and divide by the total in the universal set.
- Translate set notation into a region before you count: n(A) is the total in circle A, the intersection of A and B is the overlap only, the union of A and B is everything in either circle counted once, and the complement A' is everything outside circle A.
Worked example
In a survey of 32 students, 19 own a bicycle, 14 own a skateboard, and 8 own both a bicycle and a skateboard. (a) Draw a Venn diagram to show this information. (b) Find the number of students who own neither a bicycle nor a skateboard. (c) A student is picked at random from the 32. Find the probability that the student owns a bicycle but not a skateboard.
- Place 8 in the overlap of the two circles, since 8 students own both a bicycle and a skateboard.
- Bicycle only = 19 - 8 = 11, so write 11 in the bicycle circle outside the overlap.
- Skateboard only = 14 - 8 = 6, so write 6 in the skateboard circle outside the overlap.
- Add the three regions inside the circles: 11 + 8 + 6 = 25 students own at least one of the two items.
- Subtract from the universal set to find those outside both circles: 32 - 25 = 7 students own neither, written inside the rectangle but outside both circles.
- For part (c), 'bicycle but not skateboard' is the bicycle-only region, 11 students, so P(bicycle but not skateboard) = 11/32, which cannot be simplified further.
Practice questions
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Q1In a Year 8 class of 28 students, a Venn diagram shows two sets: G for students who play a musical instrument in a group, and S for students who play a sport for the school. There are 12 students in the overlap, 6 students in G only, and 5 students in S only. Work out how many students are in neither set.Show answer
Answer: 5 (28 - (6 + 12 + 5))
Q2A Venn diagram for 45 cafe customers shows set C (orders a coffee) and set M (orders a muffin). n(C) = 27, n(M) = 18, and 11 customers order both a coffee and a muffin. Work out how many customers order a coffee only.Show answer
Answer: 16 (27 - 11)
Q3A Venn diagram shows sets X and Y inside a rectangle labelled with a total of 50. The region for X only contains 14 elements, the region for Y only contains 21 elements, and the overlap contains 9 elements. Work out n(X or Y), the total number of elements in X or Y (or both).Show answer
Answer: 44 (14 + 21 + 9)
Q4Using set notation, describe in words what the complement of set P, written P', represents.Show answer
Answer: Every element that is not in set P (everything inside the rectangle but outside circle P).
Q5Out of 60 students, a Venn diagram shows that 22 study French only, 17 study German only, and 13 study both French and German. A student is picked at random. Find the probability that the student studies neither French nor German.Show answer
Answer: 2/15 (8/60, since 60 - (22 + 17 + 13) = 8 study neither)
Q6A Venn diagram shows 36 people at a gym: set W is people who use the weights room and set C is people who attend a class. n(W) = 24, n(C) = 19, and n(W and C) = 10. Work out n(W or C).Show answer
Answer: 33 (24 + 19 - 10)
Q7A Venn diagram for 25 dogs at a rescue centre shows set L (likes other dogs) and set K (likes cats). 16 dogs are in set L, 6 dogs are in the overlap, and 3 dogs are in neither set. Work out how many dogs are in set K only.Show answer
Answer: 6 (L only = 16 - 6 = 10; dogs in at least one set = 25 - 3 = 22; K only = 22 - 10 - 6 = 6)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A school surveyed 50 students about which clubs they attend. Let set A represent students who attend Art Club and set B represent students who attend Book Club. 21 students attend Art Club, 17 students attend Book Club, and 8 students attend both clubs. (a) Draw a Venn diagram and use it to work out how many students attend neither club. (b) One of the 50 students is picked at random. Find the probability, as a fraction in its simplest form, that the student attends Art Club but not Book Club.
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In a survey of 40 adults, set S is the set of adults who read a newspaper and set O is the set of adults who read it online. n(S) = 24, n(O) = 22, and n(S and O) = 15. Find n(S or O), and hence find how many of the 40 adults read no newspaper at all, in print or online.
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A Venn diagram for 60 members of a leisure centre shows set P (uses the pool) and set G (uses the gym). Every member uses at least one of the pool or the gym. n(P) = 38 and n(G) = 41. (a) Work out how many members use both the pool and the gym. (b) A member is picked at random. Find the probability that the member uses the gym but not the pool, giving your answer as a fraction in its simplest form.
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Free printable worksheet
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This topic is chapter 26 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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