Union, intersection and difference: a sports survey
Sixty students say whether they like football, cricket, both or neither. Change the numbers, watch the Venn diagram fill in, and find n(A ∪ B), n(A − B) and the students who like neither.
Core concept
n(A ∪ B) = n(A) + n(B) − n(A ∩ B), because the students in both are counted twice; A − B is A without B, and the complement (A ∪ B)′ is everyone in U outside both sets
Topic: Sets
In a class survey, U is the set of all 60 students asked. A is the set who like football and B the set who like cricket. A ∩ B (intersection) is the students who like both, and A ∪ B (union) is the students who like at least one of them. If you add n(A) and n(B), the students in both are counted twice, so n(A ∪ B) = n(A) + n(B) − n(A ∩ B). A − B (difference) is the students who like football but not cricket: n(A − B) = n(A) − n(A ∩ B). The complement (A ∪ B)′ is the students in U who like neither: n((A ∪ B)′) = n(U) − n(A ∪ B).
What you will learn
- Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- Find n(A − B), n(B − A) and n((A ∪ B)′) from a Venn diagram.
- Recognise disjoint sets and subsets from their numbers.
Before you start
- Know what a set, a subset and a Venn diagram are.
Materials
- A notebook to draw Venn diagrams
Equipment
- None
Procedure
- Record the survey with n(A) = 30, n(B) = 25 and n(A ∩ B) = 10.
- Make n(A ∩ B) = 0 and record: no one likes both.
- Make every football fan also a cricket fan, and record.
How this simulation is simplified
- All 60 students answered, and n(A ∩ B) cannot be more than n(A) or n(B).