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Operations on Sets · Form 4

Operations on Sets: Worked Examples (KBAT)

Real-situation problems where the set method is hidden inside the wording, a bookshop survey, event volunteers, and a three-attraction tour. Each needs you to translate the story into sets, then carry the answer one step further.

Written by the spmmath.com.my editorial team.

Worked example 1

A bookshop recorded 250 shoppers in one day. 140 bought at least one novel, 90 bought at least one stationery item, and 40 bought both a novel and stationery.

The rest bought neither. (a) How many bought neither?

(b) What percentage of all shoppers bought only a novel?

  1. Let N = bought a novel, S = bought stationery. n(N) = 140, n(S) = 90, n(N ∩ S) = 40.
  2. n(N ∪ S) = n(N) + n(S) − n(N ∩ S) = 140 + 90 − 40 = 190.
  3. (a) Bought neither = total − n(N ∪ S) = 250 − 190 = 60.
  4. Only a novel = n(N) − n(N ∩ S) = 140 − 40 = 100.
  5. (b) Percentage = (100 ÷ 250) × 100% = 40%.

Worked example 2

A residents' committee has 50 volunteers, and every volunteer helps with at least one of two events: Sports Day (S) or the Food Fair (F). 32 help with Sports Day and 27 help with the Food Fair.

Each Sports Day helper needs one blue T-shirt and each Food Fair helper needs one green T-shirt. (a) How many volunteers help with both events?

(b) How many T-shirts are needed in total?

  1. Every volunteer helps with at least one event, so no one is outside: n(S ∪ F) = 50.
  2. n(S ∪ F) = n(S) + n(F) − n(S ∩ F), so 50 = 32 + 27 − n(S ∩ F).
  3. (a) n(S ∩ F) = 59 − 50 = 9 volunteers help with both.
  4. Each Sports Day helper gets one blue shirt (32) and each Food Fair helper gets one green shirt (27); a both-event helper receives one of each.
  5. (b) Total shirts = 32 + 27 = 59.

Worked example 3

During a school trip, 80 pupils could visit three attractions: the Zoo (Z), the Museum (M) and the Aquarium (Q). The records show n(Z) = 35, n(M) = 30, n(Q) = 28, n(Z ∩ M) = 12, n(Z ∩ Q) = 10, n(M ∩ Q) = 9 and n(Z ∩ M ∩ Q) = 5.

(a) How many pupils visited exactly two attractions? (b) How many pupils visited none of the three?

  1. Start from the centre: n(Z ∩ M ∩ Q) = 5.
  2. Exactly-two regions = each pair minus the centre: Z∩M only = 12 − 5 = 7, Z∩Q only = 10 − 5 = 5, M∩Q only = 9 − 5 = 4.
  3. (a) Exactly two attractions = 7 + 5 + 4 = 16 pupils.
  4. Union by inclusion–exclusion: n(Z ∪ M ∪ Q) = 35 + 30 + 28 − 12 − 10 − 9 + 5 = 67.
  5. (b) Visited none = total − union = 80 − 67 = 13 pupils.

Worked example 4

A tour company surveyed 120 tourists about three states they visited. 60 visited Penang, 55 visited Melaka and 45 visited Langkawi.

25 visited Penang and Melaka, 20 visited Melaka and Langkawi, 15 visited Penang and Langkawi, and 10 visited all three states. Find (a) the number who visited at least one state, and (b) the number who visited none of the three states.

  1. Use the three-set formula: n(P ∪ M ∪ L) = n(P) + n(M) + n(L) − n(P∩M) − n(M∩L) − n(P∩L) + n(P∩M∩L).
  2. Substitute: 60 + 55 + 45 − 25 − 20 − 15 + 10.
  3. Add the singles: 60 + 55 + 45 = 160.
  4. Subtract the pairs: 160 − 25 − 20 − 15 = 100; then add the triple: 100 + 10 = 110.
  5. None visited = 120 − 110 = 10.

Worked example 5

In a housing area with 100 residents, 62 subscribe to Astro, 45 subscribe to Unifi, and 12 subscribe to neither service. Find (a) the number who subscribe to both services, and (b) the number who subscribe to Astro only.

  1. Residents subscribing to at least one service = 100 − 12 = 88.
  2. Use n(A ∪ U) = n(A) + n(U) − n(A ∩ U): 88 = 62 + 45 − n(A ∩ U).
  3. So n(A ∩ U) = 62 + 45 − 88 = 19.
  4. Astro only = n(A) − n(A ∩ U) = 62 − 19 = 43.

Worked example 6

A school club has 60 members. Some join the debate team and some join the drama team.

The number who join the debate team only is three times the number who join the drama team only. 12 members join both teams and 4 members join neither team.

Find the total number of members in the debate team.

  1. Let the number joining drama only be y, so debate only = 3y.
  2. Add up all four groups: (debate only) + (drama only) + (both) + (neither) = 60.
  3. 3y + y + 12 + 4 = 60, so 4y + 16 = 60.
  4. 4y = 44, giving y = 11.
  5. Debate team total = debate only + both = 3y + 12 = 3(11) + 12 = 45.

Worked example 7

At a sports carnival, 90 students signed up for at least one of two events: high jump or long jump. The number who signed up for high jump only is twice the number who signed up for long jump only.

15 students signed up for both events. Find the total number of students who signed up for high jump.

  1. Let long jump only = x, so high jump only = 2x
  2. x + 2x + 15 = 90
  3. 3x = 75
  4. x = 25, so high jump only = 2(25) = 50
  5. Total high jump = high jump only + both = 50 + 15 = 65

Worked example 8

A survey of 150 shoppers found that 80 own a smartphone, 60 own a tablet and 50 own a smartwatch. 25 own both a smartphone and a tablet, 20 own both a tablet and a smartwatch, 18 own both a smartphone and a smartwatch, and 10 own all three gadgets.

Find (a) the number who own exactly two of the three gadgets, and (b) the number who own none of the three gadgets.

  1. Exactly two = (n(A∩B) − n(A∩B∩C)) + (n(B∩C) − n(A∩B∩C)) + (n(A∩C) − n(A∩B∩C))
  2. Exactly two = (25 − 10) + (20 − 10) + (18 − 10) = 15 + 10 + 8 = 33
  3. n(A∪B∪C) = 80 + 60 + 50 − 25 − 20 − 18 + 10 = 137
  4. None of the three = 150 − 137 = 13

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

What makes KBAT questions on sets harder than standard ones?

KBAT set questions embed the maths inside a real-life scenario (say, three clubs' membership) and expect you to translate the story into a three-set Venn diagram before doing any working. The difficulty is rarely the set operations themselves, it is deciding which region a phrase like "only two of the three activities" describes.

What is the most common mistake students make with set complements in exam questions?

Students often confuse "not in A" with "not in A and not in B" when a question involves two or three sets. The complement always depends on the stated universal set, so re-check what ξ is before shading.

Also remember A′ ∩ B′ equals (A ∪ B)′, mixing up this order is a frequent KBAT trap.

What do examiners look for in a full-marks answer to an operations-on-sets question?

Marks are awarded for a correctly shaded or labelled Venn diagram, the right notation (∪, ∩, ′, ⊂), and a final answer that matches your diagram. Even if the last number is wrong, a clearly labelled diagram with correct set notation still earns partial credit, so always draw and label the diagram before calculating.

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