Operations on Sets · 4.2.2
Complement of union of sets
Students find (A ∪ B)′, the set of elements in the universal set ξ that belong to neither A nor B. This means first forming the union A ∪ B, then identifying every remaining element in ξ.
The result can be shown by shading a Venn diagram, listing elements, or using set-builder notation.
The official learning standard (4.2.2)
“Determine the complement of the union of sets.”
What it means
Students find (A ∪ B)′, the set of elements in the universal set ξ that belong to neither A nor B. This means first forming the union A ∪ B, then identifying every remaining element in ξ.
The result can be shown by shading a Venn diagram, listing elements, or using set-builder notation.
How it is examined
In Paper 1, students identify (A ∪ B)′ from a Venn diagram or listed sets, often as a multiple-choice item. In Paper 2, they shade the correct region on a Venn diagram or list the complement set explicitly, sometimes combined with other set questions in the same structured question.
Worked example
The universal set is ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Set A = {1, 2, 3, 4, 5} and set B = {4, 5, 6, 7}.
Find (A ∪ B)′.
- List the elements of A ∪ B by combining A and B without repeating elements: A ∪ B = {1, 2, 3, 4, 5, 6, 7}.
- Identify elements of ξ that are not in A ∪ B: these are 8, 9 and 10.
- So (A ∪ B)′ = {8, 9, 10}.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What's the difference between (A ∪ B)′ and A′ ∪ B′?
(A ∪ B)′ means the complement of the union, elements outside both A and B. A′ ∪ B′ combines the separate complements of A and B, which is a different (usually larger) set.
By De Morgan's law, (A ∪ B)′ = A′ ∩ B′, not A′ ∪ B′, so be careful not to mix them up.
Do I need to draw a Venn diagram every time?
Not always, if the sets are small and clearly listed, you can find A ∪ B and then the complement directly by listing. A Venn diagram helps most when sets overlap in complex ways or when the question specifically asks you to shade a region.
What if an element appears in both A and B, do I list it twice in A ∪ B?
No. A union never repeats elements, each element is listed only once even if it appears in both A and B.
Listing it twice is a common mistake that can lead to an incorrect complement, since it may cause you to miscount which elements remain in ξ.