Sets and Venn diagrams: the class sports survey
Put every student of a class in a Venn diagram of football and cricket fans, and find how many like both, one or neither.
Core concept
In short: Count a class survey with circles that overlap, without counting anyone twice.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B), because the students in both sets are counted twice
Topic: Sets and Venn diagrams
A class of 40 students is asked: do you like football? do you like cricket? Each student is a dot in the universal set U, the whole class. The football fans form set F and the cricket fans set C. Students who like both are in the overlap, the intersection F ∩ C. Everyone who likes at least one sport is in the union F ∪ C. When we add n(F) and n(C), the students in the overlap are counted twice, so n(F ∪ C) = n(F) + n(C) − n(F ∩ C). The rest of the class likes neither.
What you will learn
- Show a survey as a Venn diagram with four regions.
- Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- Find 'only A', 'only B' and 'neither' in survey problems.
Before you start
- Adding and subtracting whole numbers.
Materials
- A class survey sheet
Equipment
- Paper to draw Venn diagrams
Procedure
- Tap − or + to set the class size and how many like football, cricket and both.
- Watch every student appear as a dot in their region.
- Solve the survey: 40 students, 25 like football, 20 like cricket and 10 like both.
- Record a survey where no one likes both, one where everyone likes at least one sport, and one where every cricket fan also likes football.
How this simulation is simplified
- Every student answers both questions.