Operations on Sets · 4.3.3
Problems on combined set operations
Students apply combined set operations to solve real-life or numerical problems involving two or more overlapping groups. This typically means using given numbers of elements to find unknown quantities such as those in an intersection, a union, or a complement, often supported by a labelled Venn diagram.
The official learning standard (4.3.3)
“Solve problems involving combined operations on sets.”
What it means
Students apply combined set operations to solve real-life or numerical problems involving two or more overlapping groups. This typically means using given numbers of elements to find unknown quantities such as those in an intersection, a union, or a complement, often supported by a labelled Venn diagram.
How it is examined
This is a common Paper 2 structured question, typically a real-life scenario such as a survey on hobbies or subjects, asking students to find several unknown quantities in stages using combined operations. It may also appear as a shorter Paper 1 question testing one specific unknown value.
Worked example
A survey of 50 students found that 27 like durians, 24 like rambutans, and 11 like both fruits. Find (a) the number of students who like at least one of the two fruits, (b) the number who like neither fruit, (c) the number who like durians only.
- Let A = students who like durians, n(A) = 27; let B = students who like rambutans, n(B) = 24; n(A ∩ B) = 11.
- (a) n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 27 + 24 − 11 = 40.
- (b) Students who like neither fruit = n(ξ) − n(A ∪ B) = 50 − 40 = 10.
- (c) Students who like durians only = n(A) − n(A ∩ B) = 27 − 11 = 16.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why does 'durians only' mean A minus the intersection, not just n(A)?
n(A) includes every student who likes durians, including those who also like rambutans. 'Durians only' means students in circle A but outside the overlap with B, so you must subtract n(A ∩ B) from n(A) to exclude students who like both fruits.
What if the problem gives 'only A' and 'only B' instead of the intersection directly?
Work backwards: if you know how many like durians only and rambutans only, add those two numbers plus the overlap to get n(A ∪ B), or use the given 'only' values to first find n(A ∩ B) by subtracting from the totals n(A) and n(B).
How do I know which region of the Venn diagram represents 'neither'?
The 'neither' region is everything inside the rectangle (universal set ξ) but outside both circles A and B, it represents (A ∪ B)′. Always find n(A ∪ B) first, then subtract it from n(ξ) to get the number of students in that outer region.