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# 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.

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

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### Transcript

• 1. THE LOGIC OF COLLECTIONS Class 5 &#x2013; SetTheory andVenn Diagrams
• 2. Introduction &#xF0A7; Discrete versus Applied Mathematics &#xF0A7; Black,White &amp; Grey &#xF0FA; SetTheory andVenn &#xF0A7; Problem: Skilled Resources &#xF0FA; 24 Programmers &#xF0FA; 8 Ruby, 10 Java, 12VB &#xF0FA; 2=R+J+V &#xF0FA; 4=R+J-V &#xF0FA; 3=J+V-R &#xF0FA; 1=R-V-J &#xF0FA; ?=V-J-R Java RubyVB
• 3. Agenda &#xF0A7; Review and Debrief &#xF0A7; SetTheory &#xF0A7; Set Operators &#xF0A7; Venn Diagrams &#xF0A7; Quest: Ruby Math features &#xF0A7; QuestTopic:TruthTables &#xF0A7; Assignment &#xF0A7; Wrap-up, Questions
• 4. Review Debrief &#xF0A7; Assignment 2 &#xF0FA; Observations - Questions &#xF0A7; AssignmentThree &#xF0FA; Challenges &#xF0FA; Learning &#xF0A7; Ruby Installation &amp; IDEs &#xF0A7; Review &#x2013; Ruby Strings &amp;Variables &#xF0FA; Practice handout
• 5. Set Theory &#xF0A7; The language of Sets &#xF0FA; Set &#xF0FA; Element &#xF0FA; Subset &#xF0FA; Universe &#xF0FA; Empty set &#xF0FA; Cardinality
• 6. Set Theory &#xF0A7; Notation: Set &#xF0FA; A={1,2,3,4,5} Or: &#xF0FA; A= {x|x, a integer AND 0&lt;x&lt;6 } &#xF0FA; A={1,2,3,...,10} &#xF0FA; A={1,3,5,...,99} &#xF0FA; A={2,4,6...}
• 7. Set Theory &#xF0A7; Notation: Element &#xF0A7; x A &#xF0A7; Or A x &#xF0A7; A={1,2,3,...,10} &#xF0A7; And x= 12:. x A &#xF0FA; &#x42D; &#x42D; &#x42D;
• 8. Set Theory &#xF0A7; Notation: Subset &#xF0A7; A={1,3,5,7..99} and B = {21,27,33} &#xF0FA; B &#x2282;A &#xF0FA; Iff A&lt;&gt;B then B &#x2282;A &#xF0A7; Notation: empty set = &#xD8; or {} &#xF0FA; E={&#xD8;}; |E|=1 &#xF0FA; C= &#xD8;; |C|=0
• 9. Set Theory &#xF0A7; Notation: Universal Set &#xF0A7; Universe=U Java RubyVB U
• 10. Set Theory &#xF0A7; Notation: Cardinality &#xF0FA; A={1,3,5,...21} |A|=11 &#xF0FA; N={a,b,c,...z} |N|=26 &#xF0FA; C={1,2,3,4,...} |C|=&#x221E; &#xF0FA; Z={all even prime numbers &gt;2} |Z|=0
• 11. Exercise: Basic Set Theory &#xF0A7; Please attempt all questions &#xF0FA; Use appropriate notation &#xF0A7; Time: 10 minutes
• 12. Set Operations &#xF0A7; Union &#x2013; the set of all elements of both sets &#xF0A7; Notation: A &#x222A; B &#xF0FA; T={e,g,b,d,f} &#xF0FA; B={f,a,c,e} &#xF0FA; T &#x222A; B = {a,b,c,d,e,f,g}* A B
• 13. Set Operations &#xF0A7; Intersection &#x2013; the set of common elements of both sets &#xF0A7; Notation: A &#x2229; B &#xF0FA; T={e,g,b,d,f} &#xF0FA; B={f,a,c,e} &#xF0FA; T &#x2229; B = {e,f} A B A &#x2229; B
• 14. Set Operations &#xF0A7; Cardinality principle for two sets = &#xF0FA; |A &#x222A; B| = |A| + |B| - | A &#x2229; B | &#xF0FA; example &#xF0A7; Cardinality principle for three sets = &#xF0FA; |A &#x222A; B &#x222A; C| = |A| + |B| + |C| - |A &#x2229; B| - |A &#x2229; C| - |B &#x2229; C| + |A &#x2229; B &#x2229; C| &#xF0FA; example
• 15. Set Operations &#xF0A7; Complement &#x2013; all those elements in the universal set which are not part of the defined set &#xF0FA; Notation A&#x2019; or Ac &#xF0FA; e.g. U={1,2,3,4,...} &#xF0FA; A={2,4,6,8,...} &#xF0FA; A&#x2019;= {1,3,5,7,...}
• 16. Exercise: Set Operations &#xF0A7; Please attempt all questions &#xF0FA; Use appropriate notation &#xF0A7; Time: 10 minutes
• 17. Venn Diagrams &#xF0A7; A visual representation of Sets &#xF0A7; Each circle is a set or subset &#xF0A7; The rectangle is the universal set &#xF0A7; Overlaps are intersections &#xF0A7; The union is the set of unique elements among all sets Java RubyVB U
• 18. Venn Diagrams &#xF0A7; UsingVenn to solve problems &#xF0A7; Handout &amp; Walkthrough
• 19. Group Exercises &#xF0A7; Skills Problem &#xF0A7; LateralThinking Problem DB WEBPROG U
• 20. Group Exercises &#xF0A7; Skills Problem &#xF0A7; Plug in what we are given DB WEBPROG U=30 16 16 11 3 2 58
• 21. Group Exercises &#xF0A7; Skills Problem &#xF0A7; CalculateWEB + PROG DB WEBPROG U=30 16 16 11 3 2 58 1
• 22. Group Exercises &#xF0A7; Skills Problem &#xF0A7; Calculate PROG + DB DB WEBPROG U=30 16 16 11 3 2 58 1 4
• 23. Group Exercises &#xF0A7; Skills Problem &#xF0A7; Calculate DB 30=x + 4 + 3 + 2 + 1 + 8 + 5 &#xF0A7; 30=x + 23 &#xF0A7; X=7 DB WEBPROG U=30 16 16 11 3 2 58 1 4 x
• 24. Summary &#xF0A7; SetTheory &#xF0FA; Language and Notation &#xF0A7; Set Operations &#xF0FA; Union, Intersection, Cardinality, Complement &#xF0FA; Cardinality of two and three sets &#xF0A7; Venn Diagrams &#xF0FA; Relationship with sets &#xF0A7; Questions?
• 25. Assignment &#xF0A7; Assignment IV: SetTheory andVenn Diagrams &#xF0A7; Complete all exercise &#xF0A7; Venn and calculation required for full marks &#xF0A7; Due: Start of next class