Set Theory (Part II) Counting Principles for the Union and Intersection of Sets
In some cases, the number of elements that exist in a set is needed. With simple sets, direct counting is the quickest way.
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? = 25 Not all calculations involve ME sets
For example: Consider a group of teachers and classes. 12 math teachers 8 physics teachers 3 teach both How many teachers are there?
12: math 8: physics 3: both Can we just add them up? = 23? NO WAY!!! Try drawing a Venn Diagram
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
U Add up all the individual spaces: = 17 3 MP 95 Can we get 17 from the original numbers? =
In general: Algebraically: n(A U B) = n(A) + n(B) – n(A B) U
Consider a situation with 3 distinguishing features.
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?
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…
U T FH
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?
U T FH 4
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?
U T FH 4 8
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?
U T FH 4 8 2
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?
U T FH
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?
U T FH
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?
U T FH
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?
U T FH Add up all the numbers =48
Worksheet
Try it with our numbers The number of students involved is: – 8 – 6 – = 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
U Start by adding each subset and track the overlap … (on board)