Chapter 2: Sets, Functions and Groups

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2.1: Introduction
  1. Set Definition
  2. Problem-Set Definition
  3. Set Notation & its Characteristics
  4. Problem-Set Notation & its Characteristics
  5. Presentation of a Set
  6. Problem-Presentation of a Set
  7. Problem 1: Presentation of a Set
  8. Empty Set & Singleton Set
  9. Problem 1: Empty Set & Singleton Set
  10. Problem-Empty Set & Singleton Set
  11. Finite & Infinite Set
  12. Problem 1: Finite & Infinite Set
  13. Problem-Finite & Infinite Set
  14. Subset Definition
  15. Problem 1: Subset Definition
  16. Problem-Subset Definition
  17. Proper Subset & Improper Subset
  18. Problem 1: Proper Subset & Improper Subset
  19. Properties of Subset
  20. Problem 1: Properties of Subset
  21. Power Set
  22. Problem 1: Power Set
  23. Equal & Equivalent Sets
  24. Problem 1: Equal & Equivalent Sets
  25. Universal Set
  26. Definition of One to One Correspondence
  27. Definition & Notation of Some Important Sets
  28. Problem 1: Set Builder Notation
  29. Order of a Set
  30. Problem-Order of a Set
  31. Interval Notation of a Set
2.2: Operations on Sets
  1. Disjoint Set
  2. Overlapping Sets
  3. Union of Two Sets
  4. Problem 1: Union of Two Sets
  5. Exhaustive Sets and Cells
  6. Intersection of Set
  7. Problem 1: Intersection of Set
  8. Difference of Sets
  9. Problem 1: Difference of Sets
  10. Symmetric Difference of two sets
  11. Problem 1: Symmetric Difference of two sets
  12. Complement of a Set
  13. Problem 1: Complement of a Set
2.3: Venn Diagrams
  1. Introduction to Venn Diagrams
  2. Problem 1: Introduction to Venn Diagrams
  3. Difference of Sets for Overlapping Sets
  4. Problem 1: Difference of Sets for Overlapping Sets
  5. Difference of Sets for Disjoint Sets
  6. Problem 1: Difference of Sets for Disjoint Sets
  7. Difference of Sets for a Set & its Subset
  8. Problem 1: Difference of Sets for a Set & its Subset
  9. Complement of Union for Overlapping Sets
  10. Problem 1: Complement of Union for Overlapping Sets
  11. Complement of Intersection for Overlapping Sets
  12. Problem 1: Complement of Intersection for Overlapping Sets
  13. Complement of Intersection for Disjoint Sets
  14. Problem 1: Complement of Intersection for Disjoint Sets
  15. Complement of Union for Disjoint Sets
  16. Problem 1: Complement of Union for Disjoint Sets
  17. Complement of Union for Set & its Subset
  18. Problem 1: Complement of Union for Set & its Subset
  19. Complement of Intersection for Set & its Subset
  20. Problem 1: Complement of Intersection for Set & its Subset
  21. Demorgans Law for Overlapping Sets
  22. Problem 1: Demorgans Law for Overlapping Sets
  23. Identity Laws
  24. Problem-Identity Laws
  25. Laws of Complement
  26. Problem-Laws of Complement
  27. Some Important Laws
  28. Problem-Some Important Laws
  29. Demorgans Law for Disjoint Sets
  30. Problem 1: Demorgans Law for Disjoint Sets
  31. Demorgans Law for Set & its Subset
  32. Problem 1: Demorgans Law for Set & its Subset
2.4: Operations on Three Sets
  1. Associative Property of Union of Sets
  2. Problem 1: Associative Property of Union of Sets
  3. Associative Property of Intersection of Sets
  4. Problem 1: Associative Property of Intersection of Sets
  5. Distributive Property of Union Over Intersection
  6. Problem 1: Distributive Property of Union Over Intersection
  7. Double Complementation Law
  8. Problem-Double Complementation Law
  9. Operations of Union and Intesection on three sets
  10. Commutative Property of Union
  11. Commutativity of Intersection
  12. Distributive property of Intersection over Union
  13. Problem 1: Distributive property of Intersection over Union
  14. Distributive Property of Cartesian Product Over Union
  15. Problem-Distributive Property of Cartesian Product Over Union
  16. Distributive Property of Cartesian Product Over Intersection
  17. Problem-Distributive Property of Cartesian Product Over Intersection
  18. Distributive Property of Cartesian Product Over Difference
  19. Problem-Distributive Property of Cartesian Product Over Difference
  20. Identity With Respect to Intersection
  21. Problem-Identity With Respect to Intersection
2.5: Properties of Union and Intrersection
  1. Associative Property of Union of Sets
  2. Problem 1: Associative Property of Union of Sets
  3. Associative Property of Intersection of Sets
  4. Problem 1: Associative Property of Intersection of Sets
  5. Distributive Property of Union Over Intersection
  6. Problem 1: Distributive Property of Union Over Intersection
  7. Demorgan Law 1
  8. Problem 1: Demorgan Law 1
  9. Demorgan Law 2
  10. Problem 1: Demorgan Law 2
  11. Introduction to Venn Diagrams
  12. Problem 1: Introduction to Venn Diagrams
  13. Commutative Law of Union of Sets
  14. Problem 1: Commutative Law of Union of Sets
  15. Commutative Law of Intersection of Sets
  16. Problem 1: Commutative Law of Intersection of Sets
  17. Intersection Associativity for Overlapping Sets
  18. Problem 1: Intersection Associativity for Overlapping Sets
  19. Intersection Associativity for Disjoint Sets
  20. Problem 1: Intersection Associativity for Disjoint Sets
  21. Intersection Associativity for Sets
  22. More on Intersection Associativity for Sets
  23. Problem 1: Intersection Associativity for Sets
  24. Problem 1: More on Intersection Associativity for Sets
  25. Union Associativity for Overlapping Sets
  26. Problem 1: Union Associativity for Overlapping Sets
  27. Union Associativity for Disjoint Sets
  28. Problem 1: Union Associativity for Disjoint Sets
  29. Union Associativity for Sets
  30. Problem 1: Union Associativity for Sets
  31. More on Union Associativity for Set
  32. Problem 1: More on Union Associativity for Set
  33. Distributivity of Union Over Intersection of Sets
  34. Problem 1: Distributivity of Union Over Intersection of Sets
  35. Distributivity of Intersection Over Union of Sets
  36. Problem 1: Distributivity of Intersection Over Union of Sets
  37. Difference of Sets for Overlapping Sets
  38. Problem 1: Difference of Sets for Overlapping Sets
  39. Difference of Sets for Disjoint Sets
  40. Problem 1: Difference of Sets for Disjoint Sets
  41. Difference of Sets for a Set & its Subset
  42. Problem 1: Difference of Sets for a Set & its Subset
  43. Complement of Union for Overlapping Sets
  44. Problem 1: Complement of Union for Overlapping Sets
  45. Complement of Intersection for Overlapping Sets
  46. Problem 1: Complement of Intersection for Overlapping Sets
  47. Complement of Intersection for Disjoint Sets
  48. Problem 1: Complement of Intersection for Disjoint Sets
  49. Complement of Union for Disjoint Sets
  50. Problem 1: Complement of Union for Disjoint Sets
  51. Complement of Union for Set & its Subset
  52. Problem 1: Complement of Union for Set & its Subset
  53. Complement of Intersection for Set & its Subset
  54. Problem 1: Complement of Intersection for Set & its Subset
  55. Demorgans Law for Overlapping Sets
  56. Problem 1: Demorgans Law for Overlapping Sets
  57. Demorgans Law for Disjoint Sets
  58. Problem 1: Demorgans Law for Disjoint Sets
  59. Demorgans Law for Set & its Subset
  60. Problem 1: Demorgans Law for Set & its Subset
2.6: Inductive and Deductive Logic
  1. Concept of Induction and Deduction
  2. Problem-Concept of Induction and Deduction
  3. Aristotelian and Non Aristotelian Logics
  4. Problem-Aristotelian and Non Aristotelian Logics
  5. Symbolic Logics
  6. Problem-Symbolic Logics
2.7: Implication or Conditional
  1. Introduction to Implication or Conditional
  2. Problem-Introduction to Implication or Conditional
  3. Concept of Bi-Conditional
  4. Problem-Concept of Bi-Conditional
  5. Conditionals Related with a Given Conditional
  6. Problem-Conditionals Related with a Given Conditional
  7. Introducing Tautologies
  8. Problem-Introducing Tautologies
  9. Introducing Quantifiers
2.8: Truth Sets, A Link Between Sets Theory and Logic
  1. A link Between Set Theory and Logic
  2. More on A link Between Set Theory and Logic
  3. Further on A link Between Set Theory and Logic
  4. Problem-A link Between Set Theory and Logic
2.9: Relations
  1. Introduction to Binary Relation
  2. Problem-Introduction to Binary Relation
  3. Problem 1: Introduction to Binary Relation
  4. Domain & Range of Binary Relation
  5. Problem 1: Domain & Range of Binary Relation
  6. Problem-Domain & Range of Binary Relation
2.10: Functions
  1. Introduction to Function
  2. More on Introduction to Functions
  3. Problem 1: Introduction to Function
  4. Problem-Introduction to Function
  5. Problem3-Introduction to Function
  6. Problem4-Introduction to Function
  7. Implied Domain
  8. Finding Domain and Range of Functions
  9. Types of Relations
  10. Operations on Functions
  11. Into Function
  12. Problem 1: Into Function
  13. Problem-Into Function
  14. Onto Function
  15. Problem 1: Onto Function
  16. Problem-Onto Function
  17. One to One Function
  18. Problem 1: One to One Function
  19. Problem-One to One Correspondence
  20. Bijective Function
  21. Problem 1: Bijective Function
  22. Problem-Bijective Function
  23. Injective Function
  24. Problem 1: Injective Function
  25. Problem-Injective Function
  26. One-One Function
  27. Problem-One-One Function
  28. Set Builder Notation for a Function
  29. Problem-Set Builder Notation for a Function
  30. Formulization of Functions
  31. Algebraic Functions
  32. Linear and Quadratic Functions
  33. Problem-Linear and Quadratic Functions
  34. Sketching Graphs of Quadratic Functions
  35. Problem-Sketching Graphs of Quadratic Functions
2.11: Inverse of a Function
  1. Inverse of a Function
  2. More on Inverse of a Functions
  3. Problem1-Inverse of a Function
  4. Problem-Inverse of a Function
  5. Domain & Range of Inverse Function
  6. Square Root and Absolute Value Functions
  7. Problem-Square Root and Absolute Value Functions
  8. Constant, Identity and Rational Functions
  9. Problem-Constant, Identity and Rational Functions
  10. Even and Odd Functions
  11. Problem-Even and Odd Functions
  12. The Modulus of a Function
2.12: Binary Operations
  1. Introduction to Binary Operations
  2. Problem-Introduction to Binary Operations
  3. Multiplication Table
  4. Problem-Multiplication Table
  5. Operations on Residue Classes Modulo n
  6. Problem-Operations on Residue Classes Modulo n
  7. Properties of Binary Operations
  8. Problem-Properties of Binary Operations
  9. Theorems on Binary Operation
  10. More on Theorems on Binary Operation
  11. Problem-Theorems on Binary Operation
2.13: Groups
  1. Introduction to Semi-Group
  2. Problem-Introduction to Semi-Group
  3. Introduction to Groups
  4. Problem-Introduction to Groups
  5. Introduction to Abelian and Non-Abelian Group
  6. Problem-Introduction to Abelian and Non-Abelian Group
  7. Finite and Infinite Group
  8. Problem1-Finite and Infinite Group
  9. Problem2-Finite and Infinite Group
2.14: Solution of Linear Equations
  1. Solution of Simple Linear Equations in one Variables
  2. Problem 1: Solution of Simple Linear Equations in one Variables
  3. Problem-Solution of Linear Equations
2.15: Reversal Law of Inverse
  1. Introduction to Reversal Laws of Inverses
  2. Problem-Introduction to Reversal Laws of Inverses
  3. Theorems on Groups
  4. More on Theorems on Groups
  5. Problem-Theorems on Groups
  6. Problem-Theorems on Abelian Groups
  7. Composition of a Functions
  8. Problem-Composition of Functions
  9. Logical Proofs of the Operations on Sets
  10. More on Logical Proofs of the Operations on Sets
  11. The Set of Rational Numbers is an Abelian Group
  12. The Set of Non-Zero Real Numbers is an Abelian Group
  13. The Set of Complex Numbers is an Abelian Group
  14. Algebraic Method to find the Inverse Function
  15. Problem-Algebraic Method to find the Inverse Function
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