Three sets: a survey of newspapers read
Students were asked which of three newspapers they read. Change how many read all three, and how many were asked, and watch every region of a three-set Venn diagram change.
Core concept
For three sets, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)
Topic: Sets
In a school, students were asked which newspapers they read: A, B or C. 40 read A, 35 read B and 30 read C. 15 read both A and B, 12 read both B and C, and 10 read both C and A. Some of those read all three. If we just add 40 + 35 + 30, students who read two papers are counted twice and those who read all three are counted three times. Taking away the pairs (15 + 12 + 10) removes too much, so we add back the students in all three: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C). The students who read none are n(U) − n(A ∪ B ∪ C).
What you will learn
- Use the formula for n(A ∪ B ∪ C).
- Find the numbers in each region of a three-set Venn diagram: only one, exactly two, all three and none.
- Explain why the members of all three sets are added back.
Before you start
- Know n(A ∪ B) = n(A) + n(B) − n(A ∩ B) for two sets.
Materials
- A notebook, and coloured pencils to shade a Venn diagram
Equipment
- None
Procedure
- Record the survey with 4 students reading all three papers.
- Make it 0 students in all three and record.
- Make it 10 students in all three and record.
- Ask only 80 students and record.
How this simulation is simplified
- n(A) = 40, n(B) = 35, n(C) = 30, n(A ∩ B) = 15, n(B ∩ C) = 12 and n(C ∩ A) = 10 stay the same.