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Venn Diagrams and Set Notation - Edexcel GCSE Higher Maths

Typically 4-6 marks, usually gathered into one multi-part question · Spec 1MA1

A Venn diagram sorts things into overlapping circles drawn inside a rectangle, and that rectangle holds everything the question is about: the universal set. On Edexcel's 1MA1 specification the diagram belongs to P6, which asks you to enumerate sets and combinations of sets systematically, and it brings the notation with it: union, intersection, membership, and the complement written as a letter followed by an apostrophe. P6 is common content, so Foundation candidates fill in two-set diagrams too. What makes a question Higher is P9, conditional probability read off a diagram, along with the three-circle problems that only Higher candidates see.

Edexcel usually builds these questions the same way. One part hands you facts in words, or a universal set of numbers and two or three sets defined by a property such as 'multiples of 3', and asks you to complete the diagram, worth 2-3 marks. The next part picks a member at random and wants a probability, 1-2 marks. The Higher-only part restricts the choice, saying 'given that the student studies Physics', and that is 2 marks with a smaller denominator. It can be set on any of the three papers, and Edexcel has opened a Higher paper with one, so this is a topic you may meet in the first minute of the exam. A calculator barely helps, because the arithmetic is counting; on Paper 1 the difference is only that you leave the probability as a fraction.

Fill from the middle outwards. Most of the marks lost here go the same two ways: a set total written into the region that should hold only the members not shared, and the space outside the circles left blank when the counted regions fall short of the universal total. Add every region at the end and check the sum matches the number in the rectangle. Then read the final part twice, because 'and', 'or' and 'given that' point at three different regions, and the last of them also changes what you divide by.

Worked example

60 students were asked whether they have a part-time job and whether they play a musical instrument. 26 of the students have a part-time job. 31 of the students play a musical instrument. 12 of the students have a part-time job and play a musical instrument. One of the 60 students is chosen at random. Work out the probability that this student has neither a part-time job nor plays a musical instrument.
[4 marks]
  1. Start in the overlap. 12 students do both, so 12 goes in the region where the two circles cross.
    M1: the intersection placed first, before either single-set total is used
  2. Take the overlap off each total. Job but no instrument: 26−12=1426 - 12 = 14. Instrument but no job: 31−12=1931 - 12 = 19.
    M1: both subtractions, so no student is counted in two regions
  3. The three regions inside the circles hold 14+12+19=4514 + 12 + 19 = 45 students, so the region outside both circles holds 60−45=1560 - 45 = 15.
    M1: universal total minus the students already placed
  4. The chosen student is one of those 15 out of 60, so the probability is 1560=14\frac{15}{60} = \frac{1}{4}.
    A1: 14\frac{1}{4} or 0.25; the four regions add to 60, which is the check

Practice questions

These are original questions in Edexcel 1MA1 Higher style. Nothing here is printed as a picture, so each question either tells you the region counts in words ('7 study both') or gives you the set totals and leaves you to work the regions out. Probabilities can be typed as fractions such as 13/40, and where a question asks for a decimal it says how many decimal places.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [2 marks]
    For two sets, n(A)=17n(A) = 17, n(B)=13n(B) = 13 and n(A∪B)=24n(A \cup B) = 24. Work out n(A∩B)n(A \cap B), the number of members in both sets.
  2. Question 2 Exam pace [3 marks]
    A gym has 80 members. 46 use the swimming pool, 39 use the weights room, and 12 use neither. How many members use both the pool and the weights room?
  3. Question 3 Exam pace [3 marks]
    A cafe serves 90 breakfasts one morning. 54 of them include toast, 38 include eggs, and 17 include both toast and eggs. How many of the 90 breakfasts include neither toast nor eggs?
  4. Question 4 Exam pace [1 mark]
    FF is the set of students who study French and GG is the set of students who study German. Which expression describes the students who study French but not German?
    Answer options for question 4
  5. Question 5 Exam pace [2 marks]
    The universal set is the whole numbers from 1 to 12. Set AA is the even numbers in that range and set BB is the multiples of 3 in that range. Work out n(A∪B)n(A \cup B).
  6. Question 6 Exam pace [2 marks]
    The universal set is the whole numbers from 1 to 20. Set AA is the factors of 20 and set BB is the prime numbers in that range. Work out n(A∩B)n(A \cap B).

Common mistakes examiners see

  • Writing a whole set total into the region for that set alone. If 22 people own a dog and 9 own both a dog and a cat, putting 22 in the dog-only region makes the diagram total 9 too many, and every probability taken from it is wrong.

    Fill the overlap first, then subtract it from each set total: dog only is 22−9=1322 - 9 = 13. Finish by adding all four regions and checking the sum equals the number in the rectangle.

  • Treating the union as the overlap, so a question wanting n(A∪B)n(A \cup B) out of 80 is answered with the 21 in the intersection instead of 57.

    Union is 'or': shade circle AA, then shade circle BB, and count everything with any shading, overlap included. Intersection is 'and': only the part shaded twice. The formula n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B) subtracts the overlap because both set totals already contain it.

  • Leaving the space outside the circles empty, so a 'neither' part is answered as zero when the regions inside only account for 45 of the 60 people.

    Make the outside region the last number you write: the universal total minus everything already placed. If that subtraction gives a negative number, one of the earlier regions is too big and needs checking.

  • Dividing by the whole group on a 'given that' part, answering 12 out of 60 when the question has already narrowed the choice to the 26 students who have a part-time job.

    'Given that' shrinks the sample space. Cover everything outside the named circle with your hand, count what is still visible, and use that count as the denominator.

  • On a three-set diagram, writing n(A∩B)=28n(A \cap B) = 28 straight into the region where circles AA and BB overlap, when 9 of those 28 are in CC as well and belong in the centre.

    Put the all-three figure in the centre first, then take it off each pairwise total: the region for AA and BB but not CC holds 28−9=1928 - 9 = 19. Work outwards one ring at a time, centre, then pairs, then singles, then outside.

Frequently asked questions

Are Venn diagrams on the Foundation paper as well as Higher?
Yes. Spec point P6 covers enumerating sets and combinations of sets using tables, grids, Venn diagrams and tree diagrams, and it is common to both tiers, so Foundation candidates complete two-set diagrams and read probabilities from them. Higher adds P9, conditional probability from a diagram, and the three-circle problems where the centre has to be filled in before anything else.
What do the union and intersection symbols mean?
The union symbol is the cup-shaped one and means 'in A, or in B, or in both', so it includes the overlap and everything else inside either circle. The intersection symbol is the cap-shaped one and means only the overlap: the members that are in both sets at once. A letter followed by an apostrophe is the complement, meaning everything in the universal set that is not in that set.
How many marks are Venn diagram questions worth on Edexcel Higher?
Completing a diagram from information given in words is usually 2-3 marks, reading a probability off the finished diagram is 1-2 marks, and a conditional part is 2 marks. A whole question therefore comes to about 4-6 marks, and it can be set on Paper 1, 2 or 3.
Which set symbols do I have to know for Edexcel?
The symbols listed under P6 are the universal set, intersection, union, membership and the complement. Edexcel prints the universal set symbol beside the rectangle and tells you what that set contains, so the work is reading it rather than recalling it. Pearson also states that students are not expected to use interval bracket notation as part of set notation.

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