Class 5 - Set Theory and Venn Diagrams

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This class looks at set theory and ways to apply Venn to solving set problems.

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Class 5 - Set Theory and Venn Diagrams

  1. 1. THE LOGIC OF COLLECTIONS Class 5 – SetTheory andVenn Diagrams
  2. 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 RubyVB
  3. 3. Agenda  Review and Debrief  SetTheory  Set Operators  Venn Diagrams  Quest: Ruby Math features  QuestTopic:TruthTables  Assignment  Wrap-up, Questions
  4. 4. Review Debrief  Assignment 2  Observations - Questions  AssignmentThree  Challenges  Learning  Ruby Installation & IDEs  Review – Ruby Strings &Variables  Practice handout
  5. 5. Set Theory  The language of Sets  Set  Element  Subset  Universe  Empty set  Cardinality
  6. 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. 7. Set Theory  Notation: Element  x A  Or A x  A={1,2,3,...,10}  And x= 12:. x A  Э Э Э
  8. 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. 9. Set Theory  Notation: Universal Set  Universe=U Java RubyVB U
  10. 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. 11. Exercise: Basic Set Theory  Please attempt all questions  Use appropriate notation  Time: 10 minutes
  12. 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. 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. 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. 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. 16. Exercise: Set Operations  Please attempt all questions  Use appropriate notation  Time: 10 minutes
  17. 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 RubyVB U
  18. 18. Venn Diagrams  UsingVenn to solve problems  Handout & Walkthrough
  19. 19. Group Exercises  Skills Problem  LateralThinking Problem DB WEBPROG U
  20. 20. Group Exercises  Skills Problem  Plug in what we are given DB WEBPROG U=30 16 16 11 3 2 58
  21. 21. Group Exercises  Skills Problem  CalculateWEB + PROG DB WEBPROG U=30 16 16 11 3 2 58 1
  22. 22. Group Exercises  Skills Problem  Calculate PROG + DB DB WEBPROG U=30 16 16 11 3 2 58 1 4
  23. 23. 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 3 2 58 1 4 x
  24. 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. 25. Assignment  Assignment IV: SetTheory andVenn Diagrams  Complete all exercise  Venn and calculation required for full marks  Due: Start of next class

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