1. Set Theory (Part II) Counting Principles for the Union and Intersection of Sets

2. In some cases, the number of elements that exist in a set is needed. With simple sets, direct counting is the quickest way.

3. For example: Given any class, a student can either pass or fail. (These sets are called “mutually exclusive”) If 3 students fail, and 22 students pass, how many students are there in the class? 3 + 22 = 25 Not all calculations involve ME sets

4. For example: Consider a group of teachers and classes. 12 math teachers 8 physics teachers 3 teach both How many teachers are there?

5. 12: math 8: physics 3: both Can we just add them up? 12 + 8 + 3 = 23? NO WAY!!! Try drawing a Venn Diagram

6. U U = all the teachers in the school Begin with the overlap: 3 people like both M = math (12) P = physics (8) 3 MP 95

7. U Add up all the individual spaces: 9 + 3 + 5 = 17 3 MP 95 Can we get 17 from the original numbers? 12 8 3 17 +-=

8. In general: Algebraically: n(A U B) = n(A) + n(B) – n(A B) U

9. Consider a situation with 3 distinguishing features.

10. In a school, the following is true: 30 students are on the football team 15 are on the hockey team 25 are on the track team 8 are on football and hockey 6 are on hockey and track 12 are on football and track 4 are on all 3 teams How many students are there in total?

11. U = students at the school F = football players H = hockey players T = track members For the Venn Diagram, begin with the center and work your way out…

12. U T FH

13. In a school, the following is true: 30 students are on the football team 15 are on the hockey team 25 are on the track team 8 are on football and hockey 6 are on hockey and track 12 are on football and track 4 are on all 3 teams How many students are involved on all 3 teams?

14. U T FH 4

15. In a school, the following is true: 30 students are on the football team 15 are on the hockey team 25 are on the track team 8 are on football and hockey 6 are on hockey and track 12 are on football and track 4 are on all 3 teams How many students are involved on all 3 teams?

16. U T FH 4 8

17. In a school, the following is true: 30 students are on the football team 15 are on the hockey team 25 are on the track team 8 are on football and hockey 6 are on hockey and track 12 are on football and track 4 are on all 3 teams How many students are involved on all 3 teams?

18. U T FH 4 8 2

19. In a school, the following is true: 30 students are on the football team 15 are on the hockey team 25 are on the track team 8 are on football and hockey 6 are on hockey and track 12 are on football and track 4 are on all 3 teams How many students are involved on all 3 teams?

20. U T FH 4 8 2 4

21. In a school, the following is true: 30 students are on the football team 15 are on the hockey team 25 are on the track team 8 are on football and hockey 6 are on hockey and track 12 are on football and track 4 are on all 3 teams How many students are involved on all 3 teams?

22. U T FH 4 8 2 4 11

23. In a school, the following is true: 30 students are on the football team 15 are on the hockey team 25 are on the track team 8 are on football and hockey 6 are on hockey and track 12 are on football and track 4 are on all 3 teams How many students are involved on all 3 teams?

24. U T FH 4 8 2 4 11 5

25. In a school, the following is true: 30 students are on the football team 15 are on the hockey team 25 are on the track team 8 are on football and hockey 6 are on hockey and track 12 are on football and track 4 are on all 3 teams How many students are involved on all 3 teams?

26. U T FH 4 8 2 4 11 5 14 Add up all the numbers =48

27. Worksheet

28. Try it with our numbers The number of students involved is: 30 + 15 + 25 – 8 – 6 – 12 + 4 = 48 In general: n(A U B U C) = n(A) + n(B) + n(C) - n(A B) – n(A C) – n(B C) + n(A B C) UUU UU

29. U Start by adding each subset and track the overlap … (on board)