Class 5 - Set Theory and Venn Diagrams

The Logic of Collections
Class 5 – Set Theory and Venn Diagrams

Introduction
Discrete versus Applied Mathematics
Black, White & Grey
Set Theory and Venn
Problem: Skilled Resources
24 Programmers
8 Ruby, 10 Java, 12 VB
2=R+J+V
4=R+J-V
3=J+V-R
1=R-V-J
?=V-J-R

Agenda
Review and Debrief
Set Theory
Set Operators
Venn Diagrams
Quest: Ruby Math features
Quest Topic: Truth Tables
Assignment
Wrap-up, Questions

Review Debrief
Assignment 2
Observations - Questions
Assignment Three
Challenges
Learning
Ruby Installation & IDEs
Review – Ruby Strings & Variables
Practice handout

Set Theory
The language of Sets
Set
Element
Subset
Universe
Empty set
Cardinality

Set Theory
Notation: Set
A={1,2,3,4,5}Or:
A= {x|x, a integer AND 0<x<6 }
A={1,2,3,...,10}
A={1,3,5,...,99}
A={2,4,6...}

Set Theory
Notation: Element
x A
Or A x
A={1,2,3,...,10}
And x= 12:. x A
Э
Э
Э

Set Theory
Notation: Subset
A={1,3,5,7..99} and B = {21,27,33}
B ⊂ A
Iff A<>B then B ⊂ A
Notation: empty set = Ø or {}
E={Ø}; |E|=1
C= Ø; |C|=0

Set Theory
Notation: Universal Set
Universe=U
U

Set Theory
Notation: Cardinality
A={1,3,5,...21} |A|=11
N={a,b,c,...z} |N|=26
C={1,2,3,4,...} |C|=∞
Z={all even prime numbers >2} |Z|=0

Exercise: Basic Set Theory
Please attempt all questions
Use appropriate notation
Time: 10 minutes

Set Operations
Union – the set of all elements of both sets
Notation: A ∪ B
T={e,g,b,d,f}
B={f,a,c,e}
T ∪ B = {a,b,c,d,e,f,g}*

Set Operations
Intersection – the set of common elements of both sets
Notation: A ∩ B
T={e,g,b,d,f}
B={f,a,c,e}
T ∩ B = {e,f}
A ∩ B

Set Operations
Cardinality principle for two sets =
|A ∪ B| = |A| + |B| - | A ∩ B |
example
Cardinality principle for three sets =
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
example

Set Operations
Complement – all those elements in the universal set which are not part of the defined set
Notation A’ or Ac
e.g. U={1,2,3,4,...}
A={2,4,6,8,...}
A’= {1,3,5,7,...}

Exercise: Set Operations
Please attempt all questions
Use appropriate notation
Time: 10 minutes

Venn Diagrams
A visual representation of Sets
Each circle is a set or subset
The rectangle isthe universal set
Overlaps are intersections
The union is theset of uniqueelements among all sets
U

Venn Diagrams
Using Venn to solve problems
Handout & Walkthrough

Group Exercises
Skills Problem
Lateral ThinkingProblem
U

Group Exercises
Skills Problem
Plug in what we are given
16
U=30
2
3
16
8
5
11

Group Exercises
Skills Problem
Calculate WEB + PROG
16
U=30
2
3
1
16
8
5
11

Group Exercises
Skills Problem
Calculate PROG + DB
16
U=30
2
4
3
1
16
8
5
11

Group Exercises
Skills Problem
Calculate DB30=x + 4 + 3 + 2+ 1 + 8 + 5
30=x + 23
X=7
16
x
U=30
2
4
3
1
16
8
5
11

Summary
Set Theory
Language and Notation
Set Operations
Union, Intersection, Cardinality, Complement
Cardinality of two and three sets
Venn Diagrams
Relationship with sets
Questions?

Assignment
Assignment IV: Set Theory and Venn Diagrams
Complete all exercise
Venn and calculation required for full marks
Due: Start of next class

Class 5 - Set Theory and Venn Diagrams

by Stephen Parsons

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