THE LOGIC OF COLLECTIONSClass 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

Java

RubyVB

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

Java

RubyVB

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}*

A B

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

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 is

the universal set Overlaps are

intersections The union is the

set of uniqueelements among all sets

Java

RubyVB

U

Venn Diagrams

Using Venn to solve problems Handout & Walkthrough

Group Exercises

Skills Problem Lateral Thinking

ProblemDB

WEBPROG

U

Group Exercises

Skills Problem Plug in what we are given

DB

WEBPROG

U=30

16

16

11

32

58

Group Exercises

Skills Problem Calculate WEB + PROG

DB

WEBPROG

U=30

16

16

11

32

581

Group Exercises

Skills Problem Calculate PROG + DB

DB

WEBPROG

U=30

16

16

11

32

581

4

Group Exercises

Skills Problem Calculate DB

30=x + 4 + 3 + 2+ 1 + 8 + 5

30=x + 23 X=7

DB

WEBPROG

U=30

16

16

11

32

581

4

x

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