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Cambridge IGCSE0607

Sets

International Mathematics 0607 Chapter Notes

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Sets
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In this chapter

  1. 1Set language and notation
  2. 2Listing sets and counting elements
  3. 3Filling in and reading a Venn diagram
  4. 4Describing regions with set notation
  5. 5Elements, subsets and the empty set
  6. 6Three sets

1. Set language and notation

A set is written between curly brackets. List its elements, D = {a, b, c, …}, or give a rule: A = {x mid x is a natural number} and C = {x mid a ≤ x ≤ b}.

U is the universal set. A ∪ B is the union of A and B. A ∩ B is the intersection. A' is the complement of A.

n(A) is the number of elements in set A.

Worked example 13 marks

U = {integers from 1 to 10}, A = {x mid x is an integer, 3 ≤ x ≤ 7} and B = {2, 4, 6, 8}.

(a) List the elements of A.

(b) List the elements of A ∩ B.

(c) Find n(A').

  1. 1

    (a) 3 ≤ x ≤ 7 gives A = {3, 4, 5, 6, 7} [1]

  2. 2

    (b) in both: A ∩ B = {4, 6} [1]

  3. 3

    (c) A' is in U but not in A: A' = {1, 2, 8, 9, 10}, so n(A') = 5 [1]

2. Listing sets and counting elements

Write U as a list first. Then list each set from its rule.

For A ∪ B, list every element that is in A or in B or in both. An element in both sets is listed once.

n( ) asks for how many, so the answer is a number of elements, not a list.

Examiner insight

“Many gave 10, by counting all the elements, without taking account that 5 and 11 appear twice.” “Some did not recognise that the notation meant the number of elements so listed all the elements in A ꓴ B.”

Worked example 12 marks

A = {3, 5, 7, 9, 11, 13} and B = {5, 8, 11, 14}.

(a) List the elements of A ∩ B.

(b) Find n(A ∪ B).

  1. 1

    (a) in both sets: A ∩ B = {5, 11} [1]

  2. 2

    (b) A ∪ B = {3, 5, 7, 8, 9, 11, 13, 14}, with 5 and 11 listed once

  3. 3

    n(A ∪ B) = 8 [1]

3. Filling in and reading a Venn diagram

A Venn diagram shows sets as overlapping circles inside a rectangle for U. The regions are: A only, the overlap A ∩ B, B only, and outside both circles.

Place the elements of A ∩ B first. Then the rest of each circle. Then the elements of U in neither set. Each element goes in one region only.

Examiner insight

“The most common errors were to misplace the 1 or to omit all or some of the elements outside the two sets.” “Unfortunately, some candidates omitted this part of the question and left the Venn diagram blank.”

Worked example 12 marks

There are 24 students in a class. G = {students who study German} and S = {students who study Spanish}. The table gives the number of students in each region of the Venn diagram.

(a) Write down the number of students who do not study German and also do not study Spanish.

(b) Find n(G ∩ S').

RegionNumber of students
G only6
G and S9
S only8
neither1
  1. 1

    (a) outside both circles: 1 [1]

  2. 2

    (b) G ∩ S' is in G but not in S: the G-only region

  3. 3

    n(G ∩ S') = 6 [1]

4. Describing regions with set notation

To describe a region, name the sets by their own letters.

A ∩ B is the overlap. A ∪ B is everything inside either circle. A' is everything outside A, so A ∩ B' is the part of A that is not in B.

When a question asks for a region, make the final answer clear: give one list or one expression.

Examiner insight

“Candidates must remember to use the correct names of the sets as a few gave their answer to the second diagram as A ꓴ B.” “It is important on all questions requiring candidates to identify a region in a Venn diagram that candidates make their final answer clear.”

Worked example 13 marks

The table gives the number of elements in each region of a Venn diagram for sets A and B.

(a) Find n(A ∩ B').

(b) Find n(A ∪ B).

(c) Find n(B').

RegionNumber of elements
A only9
A and B4
B only6
neither5
  1. 1

    (a) A ∩ B' is in A but not in B: the A-only region, so n(A ∩ B') = 9 [1]

  2. 2

    (b) inside either circle: 9 + 4 + 6 = 19 [1]

  3. 3

    (c) B' is outside B: 9 + 5 = 14 [1]

5. Elements, subsets and the empty set

x ∈ A means x is an element of A. x ∉ A means it is not. A ⊆ B means A is a subset of B. A nsubseteq B means it is not.

The empty set has no elements. Write it as ∅.

A set can also hold pairs, such as B = {(x,y) mid y = mx + c}.

x ∈ A

x ∉ A

A ⊆ B

A nsubseteq B

∅

Examiner insight

“a significant number used the ‘subset’ symbol without putting set brackets round ‘Briony’.” “Some candidates wrote a few numbers for the intersection, some wrote 0, 1, 6, others wrote ‘none’ or ‘nothing’ and very few ∅.”

Worked example 13 marks

A = {2, 4, 6}, B = {x mid x is an even integer, 2 ≤ x ≤ 12} and C = {1, 2, 3}.

Complete each statement with ∈, ∉, ⊆ or nsubseteq.

(a) 4 … C

(b) A … B

(c) C … B

  1. 1

    (a) 4 is not in C, so 4 ∉ C [1]

  2. 2

    (b) B = {2, 4, 6, 8, 10, 12} and every element of A is in B, so A ⊆ B [1]

  3. 3

    (c) 1 is in C but not in B, so C nsubseteq B [1]

6. Three sets

Three circles make one region inside all three, three inside exactly two, three inside exactly one, and the outside.

Start in the middle, with A ∩ B ∩ C, and work outwards. A total such as n(A ∩ B) includes the middle.

Examiner insight

“Usually the solution was completely correct or completely incorrect due to inserting the total number in the set instead of the number in the subset, for example 25 instead of 7.”

Worked example 14 marks

36 customers are asked which of three flavours they like: A, B and C.

n(A) = 14, n(B) = 12, n(C) = 10, n(A ∩ B) = 5, n(A ∩ C) = 4, n(B ∩ C) = 3, n(A ∩ B ∩ C) = 2

Find the number who like none of the three.

  1. 1

    in all three: 2 [1]

  2. 2

    in exactly two: 5 - 2 = 3, 4 - 2 = 2, 3 - 2 = 1 [1]

  3. 3

    in exactly one: A only 14 - (2 + 3 + 2) = 7, B only 12 - (2 + 3 + 1) = 6, C only 10 - (2 + 2 + 1) = 5 [1]

  4. 4

    inside the circles: 2 + 3 + 2 + 1 + 7 + 6 + 5 = 26, so none = 36 - 26 = 10 [1]

Recap

  • n(A) is the number of elements in A. U is the universal set.
  • A ∪ B is in A or B or both. A ∩ B is in both. A' is in U but not in A.
  • Write U and each set as lists before using the notation. Count shared elements once.
  • Venn diagram: overlap first, then the rest of each circle, then outside both. One region per element.
  • A total for a set includes the overlap. A ∩ B' is in A only.
  • Extended: ∈ and ∉ compare an element with a set. ⊆ and nsubseteq compare two sets. ∅ is the empty set.
  • Extended: for three sets start in the middle and work outwards.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. A = {x mid x is a factor of 18}. (a) List the elements of A. (b) Write down n(A).2 marks
  2. Do not use a calculator. U = {integers from 1 to 12}, A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12}. (a) List A ∩ B. (b) Find n(A ∪ B). (c) List (A ∪ B)'.3 marks
  3. Do not use a calculator. U = {integers from 1 to 15}, P = {square numbers} and Q = {odd numbers}. List the elements in each region: P only, P ∩ Q, Q only, neither.4 marks
  4. 40 pupils are asked about sports. 25 play football (F). 22 play hockey (H). 6 play neither. (a) Find n(F ∩ H). (b) Find the number who play football only.3 marks
  5. A and B are two sets in U. Use set notation to describe (a) the elements in A but not in B, (b) the elements in neither A nor B, (c) the elements in B but not in A.3 marks
  6. U = {integers from 1 to 10}, E = {even numbers} and T = {multiples of 4}. Complete each statement with ∈, ∉, ⊆ or nsubseteq. (a) 8 … T (b) {8} … T (c) T … E (d) E … T4 marks
  7. Do not use a calculator. U = {integers from 20 to 30}, P = {prime numbers} and Q = {multiples of 4}. (a) Use set notation to complete the statement. P ∩ Q = (b) Find n(P ∩ Q).2 marks
  8. A school has 120 pupils. P is the set who play piano, V the set who play violin and D the set who play drums. Nobody plays both piano and violin. n(P) = 30, n(V) = 25, n(D) = 45, n(P ∩ D) = 8, n(V ∩ D) = 10 (a) Write P ∩ V in set notation and state n(P ∩ V). (b) Find the number who play none of the three. (c) Find n(D ∩ P' ∩ V').6 marks
  9. Do not use a calculator. A wildlife survey covers 60 gardens. H, F and B are the sets of gardens where a hedgehog, a fox and a badger were seen. In 8 gardens none of the three was seen. n(H) = 31, n(F) = 26, n(B) = 17, n(H ∩ F) = 12, n(H ∩ B) = 9, n(F ∩ B) = 7 (a) Find n(H ∩ F ∩ B). (b) Find the number of gardens where exactly one of the three was seen.6 marks
  10. A club has 90 members. 18 of them cannot swim. U = {members who can swim}. R, C and T are the sets of members in U who run, cycle and play tennis, and T ⊆ C. n(R) = 30, n(C) = 34, n(T) = 12, n(R ∩ C) = 11, n(R ∩ T) = 5 (a) Write down n(U). (b) Use set notation to complete the statement. T ∩ C' = (c) Find n((R ∪ C)'). (d) Find n(C ∩ R' ∩ T').7 marks

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