Operations on Sets · 4.2.3
Problems involving union of sets
Students apply the concept of union to solve real-world or numerical problems, often using the relationship n(A ∪ B) = n(A) + n(B) − n(A ∩ B). This includes counting how many items belong to at least one of several groups, interpreting Venn diagrams, and working out unknown quantities from given information.
The official learning standard (4.2.3)
“Solve problems involving the union of sets.”
What it means
Students apply the concept of union to solve real-world or numerical problems, often using the relationship n(A ∪ B) = n(A) + n(B) − n(A ∩ B). This includes counting how many items belong to at least one of several groups, interpreting Venn diagrams, and working out unknown quantities from given information.
How it is examined
This standard appears mainly in Paper 2 as multi-part structured questions involving surveys, clubs, or activities, requiring the union formula or a Venn diagram to find unknown totals. It can also appear briefly in Paper 1 as a short numerical question testing the union formula directly.
Worked example
In a class of 40 students, 25 play football and 18 play hockey. If 10 students play both games, find the number of students who play at least one of the two games.
- Let A = students who play football, so n(A) = 25. Let B = students who play hockey, so n(B) = 18.
- Given n(A ∩ B) = 10, the number who play both games.
- Use the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 25 + 18 − 10.
- n(A ∪ B) = 33.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why do we subtract n(A ∩ B) in the union formula?
If we simply add n(A) and n(B), students who belong to both groups get counted twice, once in each set. Subtracting n(A ∩ B) removes that duplicate count, so each student who plays either or both games is counted exactly once in the total.
How do I find the number who play neither game?
First find n(A ∪ B) using the formula, then subtract it from the total number of students, n(ξ). For this class, n(ξ) − n(A ∪ B) = 40 − 33 = 7 students play neither football nor hockey.
What if the problem doesn't state n(A ∩ B) directly?
Some problems describe the overlap in words, such as 'both' or 'as well as', or give it through a Venn diagram region. Read carefully to identify which number represents students counted in both groups before substituting into the union formula, since missing this step is a common source of error.