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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)
![Part IA - Numbers and Sets - SRCF · Part IA | Numbers and Sets ... [2] Sets, relations and functions Union, intersection and equality of sets. ... Modular arithmetic (congruences)](https://img.pdfslide.us/doc/110x75/5b50ad877f8b9a166e8f0888/part-ia-numbers-and-sets-srcf-part-ia-numbers-and-sets-2-sets-relations.jpg)
