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Cambridge IGCSE0580

Sets

Mathematics 0580 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 a Venn diagram
  4. 4Venn diagrams from totals
  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: x is a natural number} and C = {x: a ≤ x ≤ b}.

xi is the universal set. A ∪ B is the union of A and B. A ∩ B is the intersection: the elements in both. A' is the complement of A: the elements of xi not in A.

Examiner insight

“Some were uncertain of the symbol so the intersection and even complement symbols were seen.” “Confusion about ‘n’ standing for ‘the number in the set’ and U standing for the ‘union’ were in equal proportions.”

Worked example 13 marks

xi = {2, 4, 6, 8, 10, 12}, A = {2, 4, 6, 8} and B = {4, 8, 12}.

(a) List A ∪ B.

(b) List A ∩ B.

(c) Find n(A').

  1. 1

    (a) in A or B or both: A ∪ B = {2, 4, 6, 8, 12} [1]

  2. 2

    (b) in both: A ∩ B = {4, 8} [1]

  3. 3

    (c) A' is in xi but not in A: A' = {10, 12}, so n(A') = 2 [1]

2. Listing sets and counting elements

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

Apply the notation to the lists. For a complement, take the elements of the set out of xi. n( ) asks for how many, so the answer is a number: count the elements at the end.

Examiner insight

“Many candidates did not realise that the n(….) notation meant the number in the bracketed set.” “There were many candidates who gave incorrect answers that indicated that they did not understand the meaning of the n( ) notation.”

Worked example 12 marks

xi = {A, B, E, H, M, O, Q, R, S, U}

X = {R, H, O, M, B, U, S}

Y = {S, Q, U, A, R, E}

(i) List the elements of X'.

(ii) Find n(X ∩ Y).

  1. 1

    (i) X' is in xi but not in X: {A, E, Q} [1]

  2. 2

    (ii) in both X and Y: {R, S, U}

  3. 3

    n(X ∩ Y) = 3 [1]

3. Filling in a Venn diagram

Draw a rectangle for the universal set and one circle for each set. The circles overlap where an element is in both sets.

Put the intersection in first. Then fill the rest of each circle. Every other element of the universal set goes outside the circles. Each element is written once.

Examiner insight

“Some were confused between multiples and factors as some included multiples greater than 12 on the diagram.” “A significant number thought they only had to enter the numbers inside the circles.”

Worked example 11 mark

xi = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A = {multiples of 3}

B = {multiples of 4}

Write down n(A ∩ B).

  1. 1

    A = {3, 6, 9, 12} and B = {4, 8, 12}

  2. 2

    in both sets: A ∩ B = {12}

  3. 3

    n(A ∩ B) = 1 [1]

4. Venn diagrams from totals

A total such as n(A) = 30 counts the overlap as well. It is not the number in the A-only part.

Work inwards from the outside. Subtract the number in neither set from the total to get the number inside the circles. Add the two set totals. This counts the overlap twice, so subtract the number inside the circles to leave the overlap. Then subtract the overlap from each total.

Examiner insight

“The majority of errors in this question came from giving only the value in section C of the Venn diagram and not including the intersection.”

Worked example 14 marks

50 students are asked about languages. 30 study French (F). 25 study Spanish (S). 8 study neither.

(a) Find n(F ∩ S).

(b) Find how many study French only.

  1. 1

    (a) inside the circles: 50 - 8 = 42 [1]

  2. 2

    n(F ∩ S) = 30 + 25 - 42 [1]

  3. 3

    = 13 [1]

  4. 4

    (b) French only = 30 - 13 = 17 [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 ∅ or { }. A set can also hold pairs, such as B = {(x, y): y = mx + c}.

x ∈ A

x ∉ A

A ⊆ B

∅

Examiner insight

“Many candidates gave the incorrect answer 0.” “Some candidates used incorrect notation for the empty set, such as {Ø} or ( ) or used words, rather than set notation, such as ‘nil’, ‘null’ or ‘empty’.”

Worked example 12 marks

xi = {x: x is an integer where 40 < x < 50}

P = {x: x is a square number}

Q = {x: x is an odd number}

R = {x: x is a multiple of 3}

(a) Use set notation to complete the statement. P ∩ R =

(b) Find n(P ∩ Q ∩ R)'.

  1. 1

    P = {49} and R = {42, 45, 48}

  2. 2

    (a) 49 is not a multiple of 3, so P ∩ R = ∅ [1]

  3. 3

    (b) P ∩ Q ∩ R = ∅, so its complement is all of xi

  4. 4

    xi = {41, 42, …, 49}, so n = 9 [1]

6. Three sets

With three circles there is a region inside all three, three regions inside exactly two, three inside exactly one, and the outside.

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

Examiner insight

“The most common error was to misinterpret n(C) = 42 and to write 42 in the bottom section of the circle for set C, where they should have written 32.”

Worked example 11 mark

x is an integer.

xi = {x: 1 ≤ x ≤ 12}

A = {x: x is a square number}

B = {x: 4 ≤ x ≤ 8}

C = {x: x is a factor of 12}

Find n(A ∩ B ∩ C').

  1. 1

    A = {1, 4, 9} and B = {4, 5, 6, 7, 8}, so A ∩ B = {4}

  2. 2

    4 is a factor of 12, so 4 is in C and not in C'

  3. 3

    A ∩ B ∩ C' = ∅, so n = 0 [1]

Recap

  • n(A) is the number of elements in A. xi is the universal set.
  • A ∪ B is in A or B or both. A ∩ B is in both. A' is in xi but not in A.
  • Write the universal set and each set as lists before using the notation.
  • A Venn diagram: intersection first, then the rest of each circle, then outside.
  • A total for a set includes the overlap. Subtract the overlap to get the only-part.
  • Extended: ∅ or { } is the empty set, never 0. ∈ and ∉ compare an element with a set. ⊆ compares two sets.
  • 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: x is a prime number and x < 12} (a) List the elements of A. (b) Write down n(A).2 marks
  2. Do not use a calculator. Use set notation to write down the elements (a) in both A and B, (b) in A or B or both, (c) in xi but not in A.3 marks
  3. Do not use a calculator. xi = {x: x is an integer where 1 ≤ x ≤ 10}, A = {x: x is even} and B = {x: x is a multiple of 3}. (a) List A'. (b) Find n(A ∩ B).2 marks
  4. xi = {x: x is an integer where 20 ≤ x ≤ 30}, P = {x: x is a prime number} and Q = {x: x is a multiple of 5}. Find n(P ∪ Q)'.3 marks
  5. A club has 45 members. 28 like chess (C). 20 like draughts (D). 9 like neither. (a) Find n(C ∩ D). (b) Find n(C ∩ D').4 marks
  6. Do not use a calculator. xi = {x: x is an integer where 1 ≤ x ≤ 20}, A = {x: x is a multiple of 6} and B = {x: x is a prime number}. (a) Use set notation to complete the statement. A ∩ B = (b) Complete the statement with ∈ or ∉. 15 … B2 marks
  7. Do not use a calculator. A = {1, 2, 3}, B = {x: x is an integer where 0 < x < 6} and C = {2, 3, 6}. Complete each statement with ⊆ or nsubseteq. (a) A … B   (b) C … B2 marks
  8. 40 students are asked if they study history (H), geography (G) and music (M). n(H) = 20, n(G) = 18, n(M) = 14, n(H ∩ G) = 7, n(H ∩ M) = 5, n(G ∩ M) = 6, n(H ∩ G ∩ M) = 3 Find the number who study none of the three.5 marks
  9. Do not use a calculator. Lockers are numbered with the integers from 30 to 50 inclusive. xi = {locker numbers}, P = {prime numbers}, S = {square numbers} and T = {multiples of 6}. (a) Use set notation to complete the statement. P ∩ T = (b) Find n(P ∪ S ∪ T)'. (c) Complete the statement with ⊆ or nsubseteq. S ∩ T … P'5 marks
  10. 80 camera traps are set in a forest to record foxes (F), badgers (B) and deer (D). n(F) = 31, n(B) = 27, n(F ∩ B) = 12, n(F ∩ D) = 10, n(B ∩ D) = 9, n(F ∩ B ∩ D) = 4, n(F ∪ B ∪ D)' = 14 (a) Find the number of traps that record a deer but no fox and no badger. (b) Find n(D). (c) Find n(B ∩ (F ∪ D)').7 marks

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