This document covers set theory, Venn diagrams, and their applications. It introduces key concepts in set theory, such as sets, elements, subsets, unions, intersections, complements and cardinality. It also explains how to represent relationships between sets visually using Venn diagrams and how to solve problems using Venn diagrams. Examples are provided for set operations and using Venn diagrams to calculate values for skills problems. Students are assigned exercises to practice these set theory and Venn diagram concepts.

1. THE LOGIC OF COLLECTIONS
Class 5 – SetTheory andVenn Diagrams

2. Introduction
 Discrete versus Applied Mathematics
 Black,White & Grey
 SetTheory andVenn
 Problem: Skilled Resources
 24 Programmers
 8 Ruby, 10 Java, 12VB
 2=R+J+V
 4=R+J-V
 3=J+V-R
 1=R-V-J
 ?=V-J-R
Java
Ruby
VB

3. Agenda
 Review and Debrief
 SetTheory
 Set Operators
 Venn Diagrams
 Quest: Ruby Math features
 QuestTopic:TruthTables
 Assignment
 Wrap-up, Questions

4. Review Debrief
 Assignment 2
 Observations - Questions
 AssignmentThree
 Challenges
 Learning
 Ruby Installation & IDEs
 Review – Ruby Strings &Variables
 Practice handout

5. Set Theory
 The language of Sets
 Set
 Element
 Subset
 Universe
 Empty set
 Cardinality

6. 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...}

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

8. 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

9. Set Theory
 Notation: Universal Set
 Universe=U
Java
Ruby
VB
U

10. 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

11. Exercise: Basic Set Theory
 Please attempt all questions
 Use appropriate notation
 Time: 10 minutes

12. 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}*
A B

13. 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
A ∩ B

14. 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

15. 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,...}

16. Exercise: Set Operations
 Please attempt all questions
 Use appropriate notation
 Time: 10 minutes

17. Venn Diagrams
 A visual representation of Sets
 Each circle is a set or subset
 The rectangle is
the universal set
 Overlaps are
intersections
 The union is the
set of unique
elements among all sets
Java
Ruby
VB
U

18. Venn Diagrams
 UsingVenn to solve problems
 Handout & Walkthrough

19. Group Exercises
 Skills Problem
 LateralThinking
Problem
DB
WEB
PROG
U

20. Group Exercises
 Skills Problem
 Plug in what we are given
DB
WEB
PROG
U=30
16
16
11
3
2
5
8

21. Group Exercises
 Skills Problem
 CalculateWEB + PROG
DB
WEB
PROG
U=30
16
16
11
3
2
5
8
1

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

23. Group Exercises
 Skills Problem
 Calculate DB
30=x + 4 + 3 + 2
+ 1 + 8 + 5
 30=x + 23
 X=7
DB
WEB
PROG
U=30
16
16
11
3
2
5
8
1
4
x

24. Summary
 SetTheory
 Language and Notation
 Set Operations
 Union, Intersection, Cardinality, Complement
 Cardinality of two and three sets
 Venn Diagrams
 Relationship with sets
 Questions?

25. Assignment
 Assignment IV: SetTheory andVenn
Diagrams
 Complete all exercise
 Venn and calculation required for full marks
 Due: Start of next class