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Operations on Sets · Form 4

Operations on Sets: Worked Examples (Medium)

Two- and three-step problems that combine operations: turning a 'neither' clue into a union, writing set notation for a shaded region, and evaluating a mixed expression such as A′ ∩ B.

Written by the spmmath.com.my editorial team.

Worked example 1

A class has 30 students. 18 study Mandarin, 15 study Tamil, and 5 study neither language.

How many students study both Mandarin and Tamil?

  1. Students in at least one language = total − neither = 30 − 5 = 25, so n(M ∪ T) = 25.
  2. Use n(M ∪ T) = n(M) + n(T) − n(M ∩ T).
  3. Substitute: 25 = 18 + 15 − n(M ∩ T) = 33 − n(M ∩ T).
  4. So n(M ∩ T) = 33 − 25 = 8.

Worked example 2

In a Venn diagram of two sets A and B, the shaded region covers everything inside A and everything inside B, but the part where A and B overlap is left unshaded. Write the shaded region in set notation.

  1. 'Inside A or inside B' is the union A ∪ B.
  2. The overlap that must be removed is the intersection A ∩ B.
  3. Removing a set means intersecting with its complement, (A ∩ B)′.
  4. So the shaded region is (A ∪ B) ∩ (A ∩ B)′.

Worked example 3

The universal set is ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Set A = {multiples of 2 in ξ} and set B = {multiples of 3 in ξ}.

List the elements of A′ ∩ B.

  1. List A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12}.
  2. A′ is everything in ξ not in A: A′ = {1, 3, 5, 7, 9, 11}.
  3. A′ ∩ B keeps only the elements that are in both A′ and B.
  4. From B = {3, 6, 9, 12}, only 3 and 9 are in A′, so A′ ∩ B = {3, 9}.

Worked example 4

In a class of 40 pupils, 25 play badminton, 20 play chess and 8 play both games. Find (a) the number who play badminton only, and (b) the number who play neither game.

  1. Badminton only = 25 − 8 = 17.
  2. Number who play at least one game = 25 + 20 − 8 = 37.
  3. Neither game = total − (at least one) = 40 − 37 = 3.

Worked example 5

The diagram shows three sets A, B and C drawn inside a universal set. The shaded region lies inside both A and B but outside C.

Write the shaded region in set notation.

  1. 'Inside both A and B' means the intersection A ∩ B.
  2. 'Outside C' means the complement C′.
  3. Combine the two conditions with intersection: (A ∩ B) ∩ C′.
  4. This is usually written A ∩ B ∩ C′.

Worked example 6

Given ξ = {1, 2, 3, …, 15}, A = {prime numbers} and B = {multiples of 3}, list the elements of (A ∪ B)′.

  1. List A: prime numbers up to 15 are 2, 3, 5, 7, 11, 13.
  2. List B: multiples of 3 up to 15 are 3, 6, 9, 12, 15.
  3. A ∪ B = {2, 3, 5, 6, 7, 9, 11, 12, 13, 15}.
  4. (A ∪ B)′ = elements of ξ not in the union = {1, 4, 8, 10, 14}.

Worked example 7

In a class of 45 students, 28 like durian, 22 like mangosteen, and 6 like neither fruit. (a) How many students like both fruits?

(b) How many students like only mangosteen?

  1. Students who like at least one fruit = 45 − 6 = 39
  2. n(D ∪ M) = n(D) + n(M) − n(D ∩ M)
  3. 39 = 28 + 22 − n(D ∩ M)
  4. n(D ∩ M) = 50 − 39 = 11
  5. Only mangosteen = n(M) − n(D ∩ M) = 22 − 11 = 11

Worked example 8

Given ξ = {1, 2, 3, …, 20}, A = {factors of 20} and B = {multiples of 4 in ξ}, list the elements of A ∩ B′.

  1. A = {1, 2, 4, 5, 10, 20}
  2. B = {4, 8, 12, 16, 20}
  3. B′ = ξ \ B = {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19}
  4. A ∩ B′ = elements of A that are not in B
  5. A ∩ B′ = {1, 2, 5, 10}

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

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Frequently asked questions

How do I approach a three-set Venn diagram question without getting confused?

Fill in the innermost region (where all three sets overlap) first, then work outward to the two-set overlaps, then the parts belonging to only one set. Working from the centre out avoids double-counting, which is the most common error in three-set problems.

What's the best way to shade a region described by combined set operations?

Break the expression into steps, shade each individual set operation on a separate small Venn diagram if needed, then combine them in the order given by the brackets. Doing it in one go often causes errors when three or more sets are involved.

Why do my answers for "number of elements in a region" keep coming out wrong?

This usually happens from double-counting elements that belong to overlapping regions. Always work out the overlapping regions first using the given totals, then subtract them from single-set totals to find elements unique to just one set, checking that all regions add up to the given total.

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