Operations on Sets · 4.1.2
Complement of intersection of sets
This standard asks students to find (A ∩ B)′, every element of the universal set that does NOT belong to the intersection of A and B. Students identify this region on a Venn diagram or by listing, understanding it as the universal set minus the elements common to A and B.
The official learning standard (4.1.2)
“Determine the complement of the intersection of sets.”
What it means
This standard asks students to find (A ∩ B)′, every element of the universal set that does NOT belong to the intersection of A and B. Students identify this region on a Venn diagram or by listing, understanding it as the universal set minus the elements common to A and B.
How it is examined
In Paper 1, objective items ask students to list or shade (A ∩ B)′ given sets or a Venn diagram. In Paper 2, this may be one part of a structured question involving multiple set operations, requiring students to first find A ∩ B before determining its complement.
Worked example
Given ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 3, 5, 7} and B = {2, 4, 6, 8, 10}. Find (A ∩ B)′.
- Find A ∩ B first: the elements common to A and B are {2}.
- (A ∩ B)′ means all elements in ξ that are not in A ∩ B.
- Remove 2 from ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
- (A ∩ B)′ = {1, 3, 4, 5, 6, 7, 8, 9, 10}.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Is (A ∩ B)′ the same as A′ ∩ B′?
No, they represent different regions. (A ∩ B)′ is everything outside the overlap of A and B, which actually equals A′ ∪ B′.
A′ ∩ B′ is the smaller region outside both A and B.
What is the complement taken relative to?
The complement is always taken relative to the universal set, ξ, which must be clearly given or defined in the question. Without knowing ξ, you cannot determine (A ∩ B)′, since it depends entirely on what elements exist in the universal set.
On a Venn diagram, how do I shade (A ∩ B)′ correctly?
Shade the whole rectangle representing the universal set, then leave unshaded only the overlapping region of circles A and B. Everything shaded, including areas outside both circles, represents (A ∩ B)′.