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Venn Diagrams and Set Notation

Using sets, Venn diagrams and set notation to organise data and find probabilities.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Use set notation: \(\in\), \(\cap\), \(\cup\), the complement \(A'\), and the universal set \(\xi\).
  2. 2Draw and complete a Venn diagram for two sets from given information.
  3. 3Use a Venn diagram to work out probabilities.
  4. 4Describe regions of a Venn diagram using set notation.

Sorting into overlapping groups

Many things belong to more than one group: a student can play football and tennis, or only one, or neither. A Venn diagram shows this with overlapping circles, and it is the best way to keep track of numbers in questions that say "both", "only" and "neither". The usual trap is to forget that a number given for a whole circle includes the overlap, so the key is to start from the middle and work outwards.

Set notation

Sets are collections of items, written inside curly brackets. Learn the symbols, as questions use them without explanation.

  • Universal set \(\xi\)

    Everything being considered, drawn as the rectangle round the circles.

  • Element \(\in\)

    \(3 \in A\) means 3 is a member of set \(A\).

  • Intersection \(A \cap B\)

    The items in both \(A\) and \(B\), the overlap.

  • Union \(A \cup B\)

    The items in \(A\) or \(B\) or both, everything inside either circle.

  • Complement \(A'\)

    The items not in \(A\), everything outside the circle for \(A\).

Sets of numbers

\(\xi = \{1, 2, 3, \ldots, 10\}\), \(A = \{2, 4, 6, 8, 10\}\) and \(B = \{3, 6, 9\}\). List the members of \(A \cap B\), \(A \cup B\) and \(A'\).

Show the solutionHide the solution
  1. 1 Intersection Only 6 is in both sets, so \(A \cap B = \{6\}\).
  2. 2 Union Put every member of either set in once: \(\{2, 3, 4, 6, 8, 9, 10\}\).
  3. 3 Complement The numbers from 1 to 10 that are not even are \(1, 3, 5, 7, 9\).
  4. 4 Count check \(n(A \cup B) = 7\), which is \(5 + 3 - 1\), because 6 is counted twice if you just add.

Answer\(A \cap B = \{6\}\), \(A \cup B = \{2, 3, 4, 6, 8, 9, 10\}\), \(A' = \{1, 3, 5, 7, 9\}\)

Filling in a Venn diagram

Work from the middle outwards, one step at a time.

  • Step 1: the overlap

    Start with the number who are in both groups.

  • Step 2: the rest of each circle

    Subtract the overlap from each circle's total. If 18 play football and 6 play both, 12 play football only.

  • Step 3: the outside

    Subtract everyone inside the circles from the grand total.

  • Step 4: check

    All the regions should add up to the total.

A Venn diagram from information

30 students are asked about football (\(F\)) and tennis (\(T\)). 18 play football, 14 play tennis and 6 play both. Work out how many play neither.

Show the solutionHide the solution
  1. 1 Overlap 6 play both.
  2. 2 Football only \(18 - 6 = 12\).
  3. 3 Tennis only \(14 - 6 = 8\).
  4. 4 Neither \(30 - (12 + 6 + 8) = 30 - 26 = 4\).

Answer4

Probability from a Venn diagram

The numbers in a region are the favourable outcomes, and the total is the number in the whole rectangle.

  • One region

    \(P(F \text{ only}) = \dfrac{12}{30} = \dfrac{2}{5}\).

  • A union

    \(P(F \cup T) = \dfrac{26}{30} = \dfrac{13}{15}\).

  • A complement

    \(P(\text{neither}) = \dfrac{4}{30} = \dfrac{2}{15}\).

  • A student is chosen

    The probability that a student chosen at random is in both sets is \(\dfrac{6}{30} = \dfrac{1}{5}\).

Three sets and describing regions (Higher tier)

The same method works with a third circle, and some questions ask for the notation of a shaded region.

  • Three circles

    Fill in the middle where all three overlap first, then the regions where two overlap, then the rest.

  • Shaded regions

    The region in \(A\) but not \(B\) can be written \(A \cap B'\). Everything except the overlap is \((A \cap B)'\).

  • Addition rule

    \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).

  • Checking

    After filling in, each circle's total should match the information in the question.

Test yourself

  1. 1

    What does \(A \cap B\) mean?

    Show answerHide answer

    The items that are in both \(A\) and \(B\).

  2. 2

    What does \(A \cup B\) mean?

    Show answerHide answer

    The items that are in \(A\) or \(B\) or both.

  3. 3

    What does \(A'\) mean?

    Show answerHide answer

    The items that are not in \(A\).

  4. 4

    In a Venn diagram, where do you start filling in numbers?

    Show answerHide answer

    In the overlap.

  5. 5

    18 play football, 6 of them also play tennis. How many play football only?

    Show answerHide answer

    \(18 - 6 = 12\).

Exam technique: Venn diagrams

Be careful about totals that include the overlap.

  • Use the middle first

    Subtract the overlap from each circle before writing in the single regions.

  • Check the total

    Add every region, including the outside.

  • Read the question

    "Football" means everyone in the football circle, but "football only" excludes the overlap.

  • Write probabilities as fractions

    Use the total of the whole rectangle on the bottom.

Summary and exam focus

  • A Venn diagram uses overlapping circles inside a rectangle to show groups and their overlap.
  • \(\cap\) means both, \(\cup\) means either or both, and \(A'\) means not in \(A\).
  • Fill in the overlap first, then the rest of each circle, then the outside.
  • Probabilities from a Venn diagram use the region as the top and the whole rectangle as the bottom.

Exam focus

In a class of 40 students, 22 play football, 19 play tennis and 5 play neither. Work out the number who play both. (3 marks) (3 marks)

The number who play at least one sport is \(40 - 5 = 35\). Adding the circles gives \(22 + 19 = 41\), which counts the overlap twice, so the overlap is \(41 - 35 = 6\). A Venn diagram is a good way to show this.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Set
A collection of items, written in curly brackets.
Universal set
The set of everything being considered, written \(\xi\).
Intersection
The items in both sets, written \(A \cap B\).
Union
The items in either set or both, written \(A \cup B\).
Complement
The items not in a set, written \(A'\).
Element
A member of a set.
Venn diagram
A diagram with overlapping circles showing how sets overlap.
Overlap
The region where two circles intersect.
Region
One of the separate parts of a Venn diagram.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Write down 2 marks Easier

\(\xi = \{1, 2, 3, \ldots, 12\}\). \(A\) is the set of factors of 12 and \(B\) is the set of odd numbers. (a) Write down \(A \cap B\). [1 mark] (b) Work out \(n(A' \cap B')\). [1 mark]

Mark scheme — 2 marks available

  • (a) \(\{1, 3\}\) — B1
  • (b) 2 — B1

Model answer

(a) \(A = \{1, 2, 3, 4, 6, 12\}\) and \(B = \{1, 3, 5, 7, 9, 11\}\), so \(A \cap B = \{1, 3\}\). (b) Neither set: \(\{8, 10\}\), so \(n(A' \cap B') = 2\).

2. Exam question Calculate 4 marks Core

There are 40 students in a class. 19 have a cat and 15 have a dog. The incomplete Venn diagram shows 4 students with both and 10 with neither. (a) Complete the Venn diagram. [2 marks] (b) A student is chosen at random. Calculate the probability that the student has a dog but not a cat. [2 marks]

A Venn diagram of cats and dogs with 4 in the overlap, 10 outside both circles and the other two regions empty.

Mark scheme — 4 marks available

  • (a) 15 — B1
  • (a) 11 — B1
  • (b) \(\dfrac{11}{40}\) with the correct denominator 40 — M1
  • (b) \(\dfrac{11}{40}\) — A1

Model answer

(a) Cats only is \(19 - 4 = 15\) and dogs only is \(15 - 4 = 11\). Check: \(15 + 4 + 11 + 10 = 40\). (b) \(\dfrac{11}{40}\).

3. Exam question Calculate 3 marks Core

In a survey of 80 people, 45 read newspaper \(N\), 30 read magazine \(M\) and 15 read both. Calculate the number of people who read neither. [3 marks]

Mark scheme — 3 marks available

  • \(45 + 30 - 15\) — M1
  • 60 — A1
  • 20 — A1

Model answer

At least one: \(45 + 30 - 15 = 60\). Neither: \(80 - 60 = 20\).

4. Exam question Calculate 3 marks Core

\(A\) and \(B\) are mutually exclusive events. \(P(A) = 0.5\) and \(P(B) = 0.35\). (a) Calculate \(P(A \cup B)\). [1 mark] (b) Calculate the probability that neither event happens. [1 mark] (c) Write down \(P(A \cap B)\). [1 mark]

Mark scheme — 3 marks available

  • (a) \(0.85\) — B1
  • (b) \(0.15\) — B1
  • (c) 0 — B1

Model answer

(a) \(0.5 + 0.35 = 0.85\). (b) \(1 - 0.85 = 0.15\). (c) The events cannot both happen, so \(P(A \cap B) = 0\).

5. Exam question Calculate 4 marks Stretch

In a Venn diagram of 68 people, \(x + 5\) are in set \(A\) only, \(2x\) are in both sets, \(x\) are in set \(B\) only and \(3x\) are in neither set. (a) Calculate the value of \(x\). [2 marks] (b) A person is chosen at random. Calculate the probability that the person is in both sets. [2 marks]

Mark scheme — 4 marks available

  • (a) \(7x + 5 = 68\) — M1
  • (a) \(x = 9\) — A1
  • (b) \(\dfrac{18}{68}\) — M1
  • (b) \(\dfrac{9}{34}\) — A1

Model answer

(a) \(x + 5 + 2x + x + 3x = 68\), so \(7x + 5 = 68\) and \(x = 9\). (b) Both sets: \(2x = 18\), so \(\dfrac{18}{68} = \dfrac{9}{34}\).

6. Exam question Calculate 3 marks Core

\(\xi = \{1, 2, 3, \ldots, 20\}\). \(A\) is the set of multiples of 4 and \(B\) is the set of square numbers. (a) List the members of \(A \cap B\). [1 mark] (b) Calculate \(n(A \cup B)\). [2 marks]

Mark scheme — 3 marks available

  • (a) \(\{4, 16\}\) — B1
  • (b) \(5 + 4 - 2\) or a list of the union — M1
  • (b) 7 — A1

Model answer

(a) \(A = \{4, 8, 12, 16, 20\}\) and \(B = \{1, 4, 9, 16\}\), so \(A \cap B = \{4, 16\}\). (b) \(n(A) + n(B) - n(A \cap B) = 5 + 4 - 2 = 7\).

7. Multiple choice 1 mark Easier

What does \(A \cap B\) mean?

  1. A The items in \(A\) or \(B\)
  2. B The items not in \(A\)
  3. C The items only in \(A\)
  4. D The items in both \(A\) and \(B\) Correct

Why: \(\cap\) is the intersection, the overlap.

8. Multiple choice 1 mark Easier

What does \(A \cup B\) mean?

  1. A The items in both \(A\) and \(B\)
  2. B The items not in \(B\)
  3. C The items in \(A\) or \(B\) or both Correct
  4. D The items only in \(B\)

Why: \(\cup\) is the union, everything in either circle.

9. Multiple choice 1 mark Easier

What does \(A'\) mean?

  1. A The items in \(A\) only
  2. B The items not in \(A\) Correct
  3. C The items in both sets
  4. D The number of items in \(A\)

Why: \(A'\) is the complement of \(A\).

10. Multiple choice 1 mark Core

\(A = \{2, 4, 6, 8, 10\}\) and \(B = \{3, 6, 9\}\). What is \(A \cap B\)?

  1. A \(\{6\}\) Correct
  2. B \(\{3, 6, 9\}\)
  3. C \(\{2, 3, 4, 6, 8, 9, 10\}\)
  4. D \(\{\}\)

Why: Only 6 is in both sets.

11. Multiple choice 1 mark Easier

18 students play football and 6 of them also play tennis. How many play football only?

  1. A \(18\)
  2. B \(24\)
  3. C \(6\)
  4. D \(12\) Correct

Why: \(18 - 6 = 12\).

12. Multiple choice 1 mark Core

30 students: 18 play football, 14 play tennis and 6 play both. How many play neither?

  1. A \(2\)
  2. B \(8\)
  3. C \(4\) Correct
  4. D \(10\)

Why: \(12 + 6 + 8 = 26\) play at least one, so \(30 - 26 = 4\).

13. Multiple choice 1 mark Core

In the same survey, what is the probability that a student chosen at random plays football only?

  1. A \(\dfrac{3}{5}\)
  2. B \(\dfrac{2}{5}\) Correct
  3. C \(\dfrac{1}{5}\)
  4. D \(\dfrac{18}{30}\)

Why: \(\dfrac{12}{30} = \dfrac{2}{5}\). The fraction \(\dfrac{18}{30}\) would include those who play both.

14. Multiple choice 1 mark Stretch

In a class of 40, 22 play football, 19 play tennis and 5 play neither. How many play both?

  1. A \(6\) Correct
  2. B \(3\)
  3. C \(9\)
  4. D \(17\)

Why: \(40 - 5 = 35\) play at least one, and \(22 + 19 - 35 = 6\).

15. Multiple choice 1 mark Stretch

\(n(A) = 15\), \(n(B) = 12\) and \(n(A \cap B) = 5\). What is \(n(A \cup B)\)?

  1. A \(27\)
  2. B \(32\)
  3. C \(17\)
  4. D \(22\) Correct

Why: \(15 + 12 - 5 = 22\).