← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 1 Number · Chapter 10 · pp. 85–95

Set notation and Venn diagrams

A set is a collection with a clear rule for what belongs. Set notation gives short symbols for “is in”, “is inside”, “both” and “either”, and a Venn diagram turns those ideas into overlapping circles you can read at a glance.

Learning objectives

  1. Describe a set in words, list its elements and use ∈, ∉ and n(A).
  2. Read set-builder notation such as {x: 2 ≤ x ≤ 8}.
  3. List subsets and proper subsets, and use ⊆, ⊂ and ∅.
  4. Use the universal set ℰ and the complement A′.
  5. Read and draw Venn diagrams for A ∩ B, A ∪ B and their complements.
  6. Solve counting problems with Venn diagrams, including three sets.

Book exercises: 10.1 to 10.5 and Student assessments 1 to 3 (pp. 85–95).

Sets and elements

🔑 Key ideas

  • A set is a well-defined group of objects or symbols, written inside curly brackets { }.
  • The objects are its elements. Order does not matter, and each element is listed once.
  • A set is finite if you could finish counting it, infinite if it goes on for ever (“...” shows this).
  • n(A) is the number of elements in a finite set A.
SymbolRead it asExample
{ }the set of{2, 4, 6, 8}
∈is an element of4 ∈ {2, 4, 6, 8}
∉is not an element of5 ∉ {2, 4, 6, 8}
n(A)number of elements in An({2, 4, 6, 8}) = 4
{x: ...}the set of all x such that ...{x: 2 ≤ x ≤ 8} is every number from 2 to 8
S 2 4 6 8 5 4 ∈ S 4 is inside the circle 5 ∉ S 5 is outside it n(S) = 4: four elements inside
S = {2, 4, 6, 8} drawn as a circle. Elements go inside; anything else stays outside.

📘 Example 1 · Describe {South Africa, Namibia, Egypt, Angola, ...}

  1. Look for what every element shares: they are all countries in Africa.
  2. Describe the set: countries of Africa.
  3. Two more elements: e.g. Zimbabwe, Ghana.
  4. Finite or infinite? Africa has a fixed number of countries, so the set is finite.

📘 Example 2 · Describe B = {(x, y): y = 2x − 4} and C = {x: 2 ≤ x ≤ 8}

  1. B's elements are pairs (x, y) that fit y = 2x − 4: the points on the line y = 2x − 4.
  2. Two elements of B: x = 0 gives y = −4, so (0, −4). x = 10 gives y = 16, so (10, 16).
  3. C is every number from 2 to 8 inclusive, decimals too.
  4. Two elements of C: e.g. 5 and 6.3.

📘 Example 3 · Find n(A) for A = {x: x is an integer and 3 ≤ x ≤ 12}

  1. “Integer” means whole numbers only, and ≤ includes both ends.
  2. List them: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
  3. Count: n(A) = 10.
  4. Check: 12 − 3 + 1 = 10. ✔

✏️ Try it

  1. Describe {1, 3, 6, 10, ...} in words and give the next two elements.
  2. Find n(A) for A = {prime numbers between 50 and 60}.
  3. True or false: 7 ∈ {even numbers}?
Answers

1. Triangle numbers: 15, 21 · 2. {53, 59}, so n(A) = 2 · 3. False: 7 ∉ {even numbers}

Next: subsets →

Subsets

🔑 Key ideas

  • X ⊆ Y: X is a subset of Y. Every element of X is also in Y.
  • X ⊂ Y: X is a proper subset of Y. It is a subset, and smaller than Y.
  • ⊈ means “is not a subset of”; ⊄ means “is not a proper subset of”.
  • The empty set ∅ has no elements. It is a subset of every set.
  • Every set is a subset of itself, but not a proper subset of itself.
  • A set with n elements has 2n subsets.
J K 102030405090100 607080
K = {60, 70, 80} sits wholly inside J = {10, 20, ..., 100}, so K ⊂ J. A subset is drawn as a circle inside a circle.
start W in W out N in N out N in N out E in → {W, N, E} E out → {W, N} E in → {W, E} E out → {W} E in → {N, E} E out → {N} E in → {E} E out → ∅
Winnie, Natalie and Emma are each either in or out of a subset: 2 × 2 × 2 = 2³ = 8 subsets of {W, N, E}, from the whole set (bold) down to ∅.

💡 The book lists the proper subsets of {Winnie, Natalie, Emma} as the six in between: it leaves out the whole set (not proper) and also ∅. Many textbooks count ∅ as a proper subset too, so state what you've done.

📘 Example 4 · A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. List the subsets B {even numbers} and C {prime numbers}.

  1. Go through A and keep the even numbers: B = {2, 4, 6, 8, 10}.
  2. Go through A and keep the primes (exactly two factors; 1 is not prime): C = {2, 3, 5, 7}.
  3. Both are proper subsets of A: B ⊂ A and C ⊂ A.

📘 Example 5 · List all the subsets of A = {Winnie, Natalie, Emma}

  1. All three: {Winnie, Natalie, Emma}.
  2. Pairs: {Winnie, Natalie}, {Winnie, Emma}, {Natalie, Emma}.
  3. Singles: {Winnie}, {Natalie}, {Emma}.
  4. None: ∅.
  5. Count: 1 + 3 + 3 + 1 = 8 subsets. Check: 23 = 8. ✔

📘 Example 6 · True or false? (a) {2, 3} ⊂ {1, 2, 3} (b) {1, 2, 3} ⊂ {1, 2, 3} (c) {4} ⊆ {1, 2, 3}

  1. (a) 2 and 3 are both in {1, 2, 3}, and the set is smaller: true.
  2. (b) A set is never a proper subset of itself: false (but {1, 2, 3} ⊆ {1, 2, 3} is true).
  3. (c) 4 ∉ {1, 2, 3}, so {4} is not a subset: false.

✏️ Try it

  1. List all the subsets of {x, y}.
  2. How many subsets does a set with 4 elements have?
  3. True or false: {3, 5} ⊆ {odd numbers}?
Answers

1. ∅, {x}, {y}, {x, y} · 2. 24 = 16 · 3. True

Next: the universal set →

The universal set and the complement

🔑 Key ideas

  • The universal set ℰ holds every possible element for the problem. It is drawn as the rectangle.
  • The complement A′ is everything in ℰ that is not in A: the part of the rectangle outside A.
  • ℰ′ = ∅ and ∅′ = ℰ.
  • n(A) + n(A′) = n(ℰ).
ℰ A 12345 678910 A′ (shaded)
ℰ = {1, 2, ..., 10} and A = {1, 2, 3, 4, 5}. The shaded part outside A is A′ = {6, 7, 8, 9, 10}.

📘 Example 7 · ℰ = {1, 2, ..., 10} and A = {1, 2, 3, 4, 5}. Find A′.

  1. Start with every element of ℰ.
  2. Cross out the ones in A: 1, 2, 3, 4, 5.
  3. What is left is A′ = {6, 7, 8, 9, 10}.
  4. Check: n(A) + n(A′) = 5 + 5 = 10 = n(ℰ). ✔

📘 Example 8 · ℰ = {all 3D shapes} and P = {prisms}. Describe P′.

  1. P′ is everything in ℰ that is not in P.
  2. P′ = {all 3D shapes except prisms}, e.g. spheres, cones, pyramids.

📘 Example 9 · ℰ = {1, 2, ..., 12} and M = {multiples of 3}. Find M′ and n(M′).

  1. List M inside ℰ: M = {3, 6, 9, 12}.
  2. Take those away from ℰ: M′ = {1, 2, 4, 5, 7, 8, 10, 11}.
  3. n(M′) = 8. Check: 12 − 4 = 8. ✔

✏️ Try it

  1. ℰ = {a, b, c, d, e} and V = {a, e}. Find V′.
  2. n(ℰ) = 20 and n(A) = 7. Find n(A′).
  3. What is ℰ′?
Answers

1. {b, c, d} · 2. 20 − 7 = 13 · 3. ∅, the empty set

Next: Venn diagrams →

Venn diagrams: intersection and union

🔑 Key ideas

  • A ∩ B (intersection): elements in both A and B. It is the overlap of the circles.
  • A ∪ B (union): elements in A or B or both. List each element once.
  • Fill a Venn diagram from the middle out: overlap first, then each circle's own part, then the outside.
  • If B ⊂ A, draw B's circle wholly inside A's.
A ∩ B in both A ∪ B in A or B or both A′ not in A B′ not in B A ∩ B′ in A only A′ ∩ B in B only (A ∩ B)′ not in both (A ∪ B)′ in neither
The eight regions you will be asked to shade. In each, A is the left circle, B the right circle and the rectangle is ℰ.
P Q 391518 612 24810 P ∩ Q = {6, 12} (shaded)
P = {3, 6, 9, 12, 15, 18} and Q = {2, 4, 6, 8, 10, 12}. 6 and 12 belong to both, so they go in the overlap.

📘 Example 10 · X = {1, 3, 6, 7, 14} and Y = {3, 9, 13, 14, 18}. Draw the Venn diagram and find X ∩ Y and X ∪ Y.

  1. Find the elements in both: 3 and 14. Write them in the overlap.
  2. The rest of X (1, 6, 7) goes in the left part of X's circle.
  3. The rest of Y (9, 13, 18) goes in the right part of Y's circle.
  4. Intersection: X ∩ Y = {3, 14}.
  5. Union, each element once: X ∪ Y = {1, 3, 6, 7, 9, 13, 14, 18}.
  6. Check: n(X) + n(Y) − n(X ∩ Y) = 5 + 5 − 2 = 8 = n(X ∪ Y). ✔

📘 Example 11 · ℰ = {1, 2, ..., 10}, A = {odd numbers}, B = {prime numbers}. Find A ∩ B′ and (A ∪ B)′.

  1. List the sets: A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}.
  2. Overlap: A ∩ B = {3, 5, 7}. A only: 1, 9. B only: 2. Outside: 4, 6, 8, 10.
  3. A ∩ B′ means “in A and not in B”, the A-only part: A ∩ B′ = {1, 9}.
  4. A ∪ B = {1, 2, 3, 5, 7, 9}, so (A ∪ B)′ is the outside: (A ∪ B)′ = {4, 6, 8, 10}.
  5. Check: 2 + 3 + 1 + 4 = 10 = n(ℰ). ✔

🧪 Try it: Venn diagram shader

Type the elements, separated by commas, then pick a region to shade.

ℰ A B

    ✏️ Try it

    ℰ = {1, 2, ..., 9}, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.

    1. Find A ∩ B.
    2. Find A ∪ B.
    3. Find (A ∪ B)′ and A′ ∩ B.
    Answers

    1. {3, 4} · 2. {1, 2, 3, 4, 5, 6} · 3. (A ∪ B)′ = {7, 8, 9}; A′ ∩ B = {5, 6}

    Next: problems involving sets →

    Problems involving sets

    Now the regions hold numbers of people, not the people themselves.

    1. Neither outside first 2. In circles total − neither 3. Overlap x n(A)+n(B)−step 2 4. Only parts n(A) − x, n(B) − x
    The two-set method. Step 3 works because the overlap is counted twice when you add n(A) and n(B).

    💡 The same idea as a formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

    ℰ P C 5 Stage 1 neither first ℰ P C 5 22 − x x 20 − x Stage 2 x in the overlap ℰ P C 5 6 16 4 Stage 3 x = 16
    The book's Physics (P) and Chemistry (C) class filled in stages: outside, then the overlap as x, then the numbers.

    📘 Example 12 · 31 students: 22 study Physics, 20 study Chemistry and 5 study neither. How many study both?

    1. Put the 5 who study neither outside the circles.
    2. Students inside the circles: 31 − 5 = 26.
    3. Let x study both. Then Physics only is 22 − x and Chemistry only is 20 − x.
    4. The three parts add to 26: (22 − x) + x + (20 − x) = 26.
    5. Simplify: 42 − x = 26, so x = 16.
    6. 16 students study both. Physics only = 6, Chemistry only = 4.
    7. Check: 6 + 16 + 4 + 5 = 31. ✔

    🧪 Try it: two-set problem solver

    ℰ A B ? ? ? ?

      Three sets Extended

      Three circles make eight regions. Start at the centre (all three) and work outwards: the “two only” parts, then the “one only” parts, then the outside.

      ℰ F B S 15910523 3 3
      Example 13 filled in: Football (F), Basketball (B) and Swimming (S). The centre 3 is in all three sets.

      📘 Example 13 · 50 students: 25 play football, 20 basketball, 18 swim. F and B: 8, B and S: 6, F and S: 5, all three: 3. How many do none?

      1. Centre: all three = 3.
      2. F and B only: 8 − 3 = 5. B and S only: 6 − 3 = 3. F and S only: 5 − 3 = 2.
      3. F only: 25 − 5 − 2 − 3 = 15.
      4. B only: 20 − 5 − 3 − 3 = 9.
      5. S only: 18 − 2 − 3 − 3 = 10.
      6. Inside the circles: 3 + 5 + 3 + 2 + 15 + 9 + 10 = 47, so n(F ∪ B ∪ S) = 47.
      7. None: 50 − 47 = 3 students.
      8. Check with the formula: 25 + 20 + 18 − 8 − 6 − 5 + 3 = 47. ✔

      ✏️ Try it

      1. 40 students: 25 like tea, 20 like coffee and 5 like neither. How many like both?
      2. In question 1, how many like tea only?
      3. n(A) = 12, n(B) = 9 and n(A ∩ B) = 4. Find n(A ∪ B).
      Answers

      1. Inside circles 35; 25 + 20 − 35 = 10 · 2. 25 − 10 = 15 · 3. 12 + 9 − 4 = 17

      Next: summary and quiz →

      Chapter summary

      • A set is a well-defined group of elements in { }. ∈ is “is an element of”, ∉ is “is not”, n(A) counts the elements.
      • {x: ...} means “all x such that ...”. Check whether x must be an integer.
      • X ⊆ Y: every element of X is in Y. X ⊂ Y: a proper subset, smaller than Y. ∅ is a subset of every set.
      • A set with n elements has 2n subsets.
      • ℰ is the universal set (the rectangle). A′ is everything in ℰ not in A.
      • A ∩ B is the overlap (both). A ∪ B is everything in either circle (or both).
      • Counting problems: neither first, then x in the overlap. n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
      • Three sets: fill the centre first and work outwards.
      SymbolMeaning
      ∈ / ∉is / is not an element of
      n(A)number of elements in A
      ⊆ / ⊈is / is not a subset of
      ⊂ / ⊄is / is not a proper subset of
      ∅the empty set
      ℰthe universal set
      A′the complement of A
      A ∩ Bintersection: in both
      A ∪ Bunion: in either or both

      📄 Practice questions

      The book's Exercises 10.1 to 10.5 and Student assessments 1 to 3 are in the printable assignment for this chapter.

      Mini-quiz

      1. A = {3, 5, 7, 9}. What is n(A)?
      2. Which statement is true for A = {2, 4, 6}?
      3. How many integers are in {x: x is an integer and 4 ≤ x ≤ 9}?
      4. How many subsets does {a, b, c, d} have?
      5. Which statement is false?
      6. ℰ = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}. What is A′?
      7. P = {1, 2, 3, 4} and Q = {3, 4, 5}. What is P ∩ Q?
      8. P = {1, 2, 3, 4} and Q = {3, 4, 5}. What is P ∪ Q?
      9. On a two-circle Venn diagram, which region is A ∩ B′?
      10. What does (A ∪ B)′ shade on a Venn diagram?
      11. 30 students: 18 do art, 15 do music, 4 do neither. How many do both?
      12. n(A) = 14, n(B) = 10 and n(A ∩ B) = 6. What is n(A ∪ B)?

      Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 10 (Set notation and Venn diagrams). Explanations, the Try it answers, the tools and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.