set theory (part ii) counting principles for the union and intersection of sets
TRANSCRIPT
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?
3 + 22 = 25
Not all calculations involve ME sets
For example: Consider a group of teachers and classes.
12 math teachers8 physics teachers
3 teach bothHow many teachers are there?
12: math8: physics
3: both
Can we just add them up? 12 + 8 + 3 = 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
M P
9 5
U
Add up all the individual spaces:
9 + 3 + 5 = 17
3
M P
9 5
Can we get 17 from the original numbers?
12 8 3 17+ - =
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 schoolF = football playersH = hockey playersT = track members
For the Venn Diagram, begin with the center and work your way out…
U
T
F H
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
F H
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
F H
48
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
F H
48 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
F H
48 2
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
F H
48 2
4
11
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
F H
48 2
4
11
5
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
F H
48 2
4
11
514
Add up all the numbers = 48
Worksheet
Try it with our numbersThe 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)
U U U
U U
U
Start by adding each subset and track the overlap … (on board)