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.
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
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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
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,...}
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
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