EN
Book a Trial Class

Operations on Sets · Form 4

Operations on Sets: Worked Examples (Easier)

Practises the three core set operations one at a time, intersection, union and complement, plus reading a simple two-set Venn diagram. Good for building confidence before combining operations.

Written by the spmmath.com.my editorial team.

Worked example 1

The universal set is ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, with A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. List the elements of A ∩ B and A ∪ B.

  1. A ∩ B means the elements found in both A and B, check each element of A against B.
  2. 2 and 4 appear in both sets, so A ∩ B = {2, 4}.
  3. A ∪ B means every element that is in A or in B (or both), listed once each.
  4. Combine {1, 2, 3, 4, 5} with {2, 4, 6, 8}, writing shared elements only once: A ∪ B = {1, 2, 3, 4, 5, 6, 8}.

Worked example 2

Given the universal set ξ = {x : 1 ≤ x ≤ 10, x is an integer} and P = {even numbers in ξ}, list the elements of the complement P′.

  1. First write out ξ in full: ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
  2. List P, the even numbers: P = {2, 4, 6, 8, 10}.
  3. P′ is everything in ξ that is not in P.
  4. Remove the even numbers, leaving P′ = {1, 3, 5, 7, 9}.

Worked example 3

In a two-set Venn diagram, the region for A only shows 7, the overlap A ∩ B shows 3, the region for B only shows 5, and the region outside both sets shows 4. Find n(A ∪ B) and n(ξ).

  1. n(A ∪ B) counts everything inside the two circles: A only + overlap + B only.
  2. n(A ∪ B) = 7 + 3 + 5 = 15.
  3. n(ξ) counts everyone, so add the outside region as well.
  4. n(ξ) = 15 + 4 = 19.

Worked example 4

Given ξ = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, list the elements of (A ∩ B)′.

  1. Find A ∩ B first: the elements common to A and B are 3 and 4, so A ∩ B = {3, 4}.
  2. The complement (A ∩ B)′ is every element of ξ that is NOT in A ∩ B.
  3. Remove 3 and 4 from ξ = {1, 2, 3, 4, 5, 6, 7, 8}.
  4. (A ∩ B)′ = {1, 2, 5, 6, 7, 8}.

Worked example 5

Set A and set B are such that n(A) = 12, n(B) = 8 and n(A ∩ B) = 5. Find n(A ∪ B).

  1. Use the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
  2. Substitute the values: n(A ∪ B) = 12 + 8 − 5.
  3. Add and subtract: 20 − 5 = 15.

Worked example 6

Given ξ = {factors of 18}, A = {even factors of 18} and B = {odd factors of 18}, list A ∩ B and A ∪ B.

  1. List ξ: the factors of 18 are 1, 2, 3, 6, 9 and 18.
  2. A = even factors = {2, 6, 18}; B = odd factors = {1, 3, 9}.
  3. A ∩ B: no factor is both even and odd, so A ∩ B = { } (the empty set).
  4. A ∪ B: combine all elements = {1, 2, 3, 6, 9, 18}.

Worked example 7

The universal set ξ has 25 elements. Set A has 15 elements.

Find n(A′).

  1. n(A′) = n(ξ) − n(A)
  2. n(A′) = 25 − 15
  3. n(A′) = 10

Worked example 8

Given ξ = {1, 2, 3, …, 12}, C = {multiples of 3 in ξ} and D = {multiples of 4 in ξ}, list the elements of C ∩ D and C ∪ D.

  1. C = {3, 6, 9, 12}
  2. D = {4, 8, 12}
  3. C ∩ D = {12}
  4. C ∪ D = {3, 4, 6, 8, 9, 12}

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

Book a Trial ClassOne-hour paid trial · Same-day reply · from RM50/hr

Frequently asked questions

What's the difference between a subset symbol and an element symbol?

"∈" means an item belongs to a set (used for a single element), while "⊆" means one whole set belongs inside another set. Mixing these up is one of the most common mistakes, check whether you're comparing an item to a set, or a set to a set.

How do I know when to use a Venn diagram versus set notation?

Venn diagrams are useful for visualising and counting overlaps between sets, while set notation (∩, ∪, ′) is used for writing exact answers. Many questions expect both, draw the diagram first to see the regions clearly, then write the matching notation.

What's the easiest way to avoid mistakes with the complement of a set?

Remember the complement (A′) contains everything in the universal set that is NOT in A. A common error is forgetting to check the universal set given in the question, so always confirm what the universal set is before finding any complement.

Book a Trial Class

Book a Trial Class
One-hour paid trial · Same-day replyfrom RM50/hr
Book a Trial Class