Video Converter
Audio Converter
Image Converter
PDF Converter
File Compressor
induction

Induction Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech - PowerPoint PPT Presentation

Oct 06, 2023 •31 likes •1.91k views

The Principle of Induction Examples Strong Induction Associativity Induction Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Induction The Principle of Induction Examples Strong

  1. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Let P ( n ) be a statement about the natural number n. If P ( 1 ) is true and if, for all n ∈ N , truth of P ( n ) implies truth of P ( n + 1 ) , then P ( n ) holds for all natural numbers. � � Proof. Consider the set S : = n ∈ N : P ( n ) is true . Then 1 ∈ S . Moreover, if n ∈ S , then the statement P ( n ) is true. By assumption, this implies that P ( n + 1 ) is true. That is, n + 1 ∈ S . By the Principle of Induction, S = N . Thus for all n ∈ N the statement P ( n ) is true. The proof of P ( 1 ) is also called the base step of the induction. The proof that P ( n ) implies P ( n + 1 ) is also called the induction step . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  2. The Principle of Induction Examples Strong Induction Associativity Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  3. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  4. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  5. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  6. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . 1 = 1 ( 1 + 1 ) . 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  7. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . 1 = 1 ( 1 + 1 ) . 2 Induction step n → n + 1 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  8. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . 1 = 1 ( 1 + 1 ) . 2 Induction step n → n + 1 . 1 + 2 + 3 + ··· + n +( n + 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  9. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . 1 = 1 ( 1 + 1 ) . 2 Induction step n → n + 1 . n ( n + 1 ) 1 + 2 + 3 + ··· + n +( n + 1 ) = +( n + 1 ) 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  10. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . 1 = 1 ( 1 + 1 ) . 2 Induction step n → n + 1 . n ( n + 1 ) 1 + 2 + 3 + ··· + n +( n + 1 ) = +( n + 1 ) 2 n ( n + 1 )+ 2 ( n + 1 ) = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  11. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . 1 = 1 ( 1 + 1 ) . 2 Induction step n → n + 1 . n ( n + 1 ) 1 + 2 + 3 + ··· + n +( n + 1 ) = +( n + 1 ) 2 n ( n + 1 )+ 2 ( n + 1 ) = 2 � � ( n + 1 ) ( n + 1 )+ 1 = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  12. The Principle of Induction Examples Strong Induction Associativity Example. For all n ∈ N we have that 1 + 2 + 3 + ··· + n = n ( n + 1 ) . 2 Base step, n = 1 . 1 = 1 ( 1 + 1 ) . 2 Induction step n → n + 1 . n ( n + 1 ) 1 + 2 + 3 + ··· + n +( n + 1 ) = +( n + 1 ) 2 n ( n + 1 )+ 2 ( n + 1 ) = 2 � � ( n + 1 ) ( n + 1 )+ 1 = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  13. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  14. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Base step logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  15. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Base step, n = 3 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  16. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Base step, n = 3 . (See book. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  17. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Base step, n = 3 . (See book. We also did quadrilaterals in the presentation on proofs.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  18. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  19. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  20. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Finding the right corner. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  21. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Finding the right corner. ✑✑✑✑✑✑ ☞ ☞ ☞ ☞ � ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  22. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Finding the right corner. A k ✑✑✑✑✑✑ s ☞ ☞ ☞ ☞ � ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  23. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Finding the right corner. A k ✑✑✑✑✑✑ s A k − 1 s ☞ ☞ ☞ ☞ � ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  24. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Finding the right corner. A k ✑✑✑✑✑✑ s A k − 1 s ☞ ☞ ☞ ☞ s � A k + 1 ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  25. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Finding the right corner. A k ✑✑✑✑✑✑ s A k − 1 s ☞ ☞ ☞ ☞ s � A k + 1 ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  26. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Finding the right corner. A k ✑✑✑✑✑✑ s A k − 1 A j s ☞ ✑✑✑ s ☞ A j − 1 ☞ s ☞ s s � A j + 1 A k + 1 ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  27. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “convex corner”. A k ✑✑✑✑✑✑ s A k − 1 A j s ☞ ✑✑✑ s ☞ A j − 1 ☞ s ☞ s s � A j + 1 A k + 1 ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  28. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “convex corner”. A k ✑✑✑✑✑✑ s A k − 1 s ☞ ☞ ☞ ☞ s � A k + 1 ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  29. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “convex corner”. A k ✑✑✑✑✑✑ s A k − 1 s ☞ α k − 1 ☞ ☞ ☞ s � A k + 1 ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  30. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “convex corner”. A k ✑✑✑✑✑✑ s α k A k − 1 s ☞ α k − 1 ☞ ☞ ☞ s � A k + 1 ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  31. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “convex corner”. A k ✑✑✑✑✑✑ s α k A k − 1 s ☞ α k − 1 ☞ ☞ ☞ α k + 1 s � A k + 1 ... � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  32. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  33. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. A k ✑✑✑✑✑✑ s s A k − 1 s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  34. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. A k ✆✆ ✑✑✑✑✑✑ s ✆ ✆ ✆ s A k − 1 s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  35. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ A k ✆✆ ✑✑✑✑✑✑ s ✆ ✆ ✆ s A k − 1 s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  36. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ A k ✆✆ ✑✑✑✑✑✑ s ✆ ✔ ✆ ✔ ✆ ✔ s ✔ A k − 1 ✔ ✔ ✔ ✔ s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  37. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ ... A k ✆✆ ✑✑✑✑✑✑ s ✆ ✔ ✆ ✔ ✆ ✔ s ✔ A k − 1 ✔ ✔ ✔ ✔ s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  38. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ ... A k ✆✆ ✑✑✑✑✑✑ s ✆ ✔ ✆ ✔ ✆ ✔ s ✔ A k − 1 ✔ ✔ ✔ ✔ s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  39. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ ... A k ✆✆ ✑✑✑✑✑✑ s ✆ ✔ ✆ ✔ ✆ ✔ s ✔ A k − 1 ✔ ✔ ✔ ✔ s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  40. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ ... A k ✆✆ ✑✑✑✑✑✑ s ✆ ✔ ✆ ✔ ✆ ✔ s ✔ A k − 1 ✔ ✔ ✔ ✔ s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  41. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ ... A k ✆✆ ✑✑✑✑✑✑ s ✆ ✔ ✆ ✔ ✆ ✔ s α k − 1 ✔ A k − 1 ✔ ✔ ✔ ✔ s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  42. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ ... A k ✆✆ ✑✑✑✑✑✑ s ✆ α k ✔ ✆ ✔ ✆ ✔ s α k − 1 ✔ A k − 1 ✔ ✔ ✔ ✔ s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  43. The Principle of Induction Examples Strong Induction Associativity Example. The sum of the interior angles of a plane n-gon with n ≥ 3 vertices is ( n − 2 ) π . Induction step. Cutting a “concave corner”. ✘ ✘✘✘✘✘✘✘✘✘ ... A k ✆✆ ✑✑✑✑✑✑ s ✆ α k ✔ ✆ ✔ ✆ ✔ s α k − 1 ✔ A k − 1 ✔ ✔ α k + 1 ✔ ✔ s A k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  44. The Principle of Induction Examples Strong Induction Associativity Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  45. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  46. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  47. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  48. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  49. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  50. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  51. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  52. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  53. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  54. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  55. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  56. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  57. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  58. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  59. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  60. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. n + 1 is the smallest element of N \{ 1 ,..., n } . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  61. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. n + 1 is the smallest element of N \{ 1 ,..., n } . Thus n + 1 �∈ A . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  62. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. n + 1 is the smallest element of N \{ 1 ,..., n } . Thus n + 1 �∈ A . Hence P ( n + 1 ) is true. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  63. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. n + 1 is the smallest element of N \{ 1 ,..., n } . Thus n + 1 �∈ A . Hence P ( n + 1 ) is true. So P ( n ) holds for all n ∈ N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  64. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. n + 1 is the smallest element of N \{ 1 ,..., n } . Thus n + 1 �∈ A . Hence P ( n + 1 ) is true. So P ( n ) holds for all n ∈ N and A is empty logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  65. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. n + 1 is the smallest element of N \{ 1 ,..., n } . Thus n + 1 �∈ A . Hence P ( n + 1 ) is true. So P ( n ) holds for all n ∈ N and A is empty, contradiction. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  66. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. n + 1 is the smallest element of N \{ 1 ,..., n } . Thus n + 1 �∈ A . Hence P ( n + 1 ) is true. So P ( n ) holds for all n ∈ N and A is empty, contradiction. Therefore A must have a smallest element. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  67. The Principle of Induction Examples Strong Induction Associativity Theorem. Principle of Induction. Every nonempty subset A of N has a smallest element. Proof. Let / 0 � = A ⊆ N . Suppose for a contradiction that A does not have a smallest element. Let P ( n ) = “ { 1 ,..., n }∩ A = / 0”. We will prove P ( n ) for all n ∈ N . Base step, n = 1 . 1 is the smallest element of N . So 1 �∈ A . Hence P ( 1 ) is true. Induction step, n → n + 1 . Assume { 1 ,..., n }∩ A = / 0 and consider the element n + 1. n + 1 is the smallest element of N \{ 1 ,..., n } . Thus n + 1 �∈ A . Hence P ( n + 1 ) is true. So P ( n ) holds for all n ∈ N and A is empty, contradiction. Therefore A must have a smallest element. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  68. The Principle of Induction Examples Strong Induction Associativity Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  69. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  70. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. Sets that are not finite are called infinite . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  71. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. Sets that are not finite are called infinite . For finite sets F � = / 0 we set | F | : = n with n as above logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  72. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. Sets that are not finite are called infinite . For finite sets F � = / 0 we set | F | : = n with n as above and we set | / 0 | : = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  73. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. Sets that are not finite are called infinite . For finite sets F � = / 0 we set | F | : = n with n as above and we set | / 0 | : = 0 , where 0 is an element that is not in N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  74. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. Sets that are not finite are called infinite . For finite sets F � = / 0 we set | F | : = n with n as above and we set | / 0 | : = 0 , where 0 is an element that is not in N . For infinite sets I we set | I | : = ∞ , where ∞ is an element that is not in N ∪{ 0 } . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  75. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. Sets that are not finite are called infinite . For finite sets F � = / 0 we set | F | : = n with n as above and we set | / 0 | : = 0 , where 0 is an element that is not in N . For infinite sets I we set | I | : = ∞ , where ∞ is an element that is not in N ∪{ 0 } . Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  76. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. Sets that are not finite are called infinite . For finite sets F � = / 0 we set | F | : = n with n as above and we set | / 0 | : = 0 , where 0 is an element that is not in N . For infinite sets I we set | I | : = ∞ , where ∞ is an element that is not in N ∪{ 0 } . Theorem. Let A and B be finite sets so that A ⊆ B and so that | A | = | B | . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  77. The Principle of Induction Examples Strong Induction Associativity Definition. A set F is called finite iff F is empty or there is an n ∈ N and a bijective function f : { 1 ,..., n } → F. Sets that are not finite are called infinite . For finite sets F � = / 0 we set | F | : = n with n as above and we set | / 0 | : = 0 , where 0 is an element that is not in N . For infinite sets I we set | I | : = ∞ , where ∞ is an element that is not in N ∪{ 0 } . Theorem. Let A and B be finite sets so that A ⊆ B and so that | A | = | B | . Then A = B. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  78. The Principle of Induction Examples Strong Induction Associativity Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

  79. The Principle of Induction Examples Strong Induction Associativity Proof. If | A | = 0, then | B | = | A | = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Induction

Recommend

Induction Stepwise induction (for T PA , T cons ) Complete induction (for T PA , T cons )

Induction Stepwise induction (for T PA , T cons ) Complete induction (for T PA , T cons )

Induction Stepwise induction (for T PA , T cons ) Complete induction (for T PA , T cons ) 4. Induction Theoretically equivalent in power to stepwise induction, but sometimes produces more concise proof Well-founded induction

383 views • 7 slides
Induction and recursion Chapter 5 Chapter Summary Mathematical Induction Strong Induction

Induction and recursion Chapter 5 Chapter Summary Mathematical Induction Strong Induction

Induction and recursion Chapter 5 Chapter Summary Mathematical Induction Strong Induction Well-Ordering Recursive Definitions Structural Induction Recursive Algorithms Mathematical Induction Section 5.1 Section Summary Mathematical

1.28k views • 75 slides
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach the first rung of the

450 views • 31 slides
MA THEMA TICAL INDUCTION Induction and Deduction Mathematical Induction (its

MA THEMA TICAL INDUCTION Induction and Deduction Mathematical Induction (its

MA THEMA TICAL INDUCTION Induction and Deduction Mathematical Induction (its strength) Examples { Sum of rst n naturual numb ers Prove b y induction that the sum of the rst n natural numb ers is

760 views • 34 slides
Beyond Inductive Definitions Induction-Recursion, Induction-Induction, Coalgebras Anton

Beyond Inductive Definitions Induction-Recursion, Induction-Induction, Coalgebras Anton

Beyond Inductive Definitions Induction-Recursion, Induction-Induction, Coalgebras Anton Setzer Swansea University, Swansea UK 1 March 2012 1/ 26 A Proof Theoretic Programme Sets in Martin-L of Type Theory Induction-Recursion,

301 views • 26 slides
Lecture Outline Strengthening Induction Hypothesis. Lecture Outline Strengthening Induction

Lecture Outline Strengthening Induction Hypothesis. Lecture Outline Strengthening Induction

Lecture Outline Strengthening Induction Hypothesis. Lecture Outline Strengthening Induction Hypothesis. Strong Induction Lecture Outline Strengthening Induction Hypothesis. Strong Induction Well ordered principle. Tutoring Option. How does

1.73k views • 154 slides
Strong induction (3) 23/38 Let P be a unary predicate on N Strong induction: Induction . . .

Strong induction (3) 23/38 Let P be a unary predicate on N Strong induction: Induction . . .

Strong induction (3) 23/38 Let P be a unary predicate on N Strong induction: Induction . . . Lecture 12 (Chapter 19) ( ) k [ k N : j [ j N j < k : P ( j )] P ( k )] { Strong induction on ( ) : } ( m ) n

88 views • 4 slides
Induction and Recursion CMPS/MATH 2170: Discrete Mathematics Outline Mathematical induction

Induction and Recursion CMPS/MATH 2170: Discrete Mathematics Outline Mathematical induction

Induction and Recursion CMPS/MATH 2170: Discrete Mathematics Outline Mathematical induction (5.1) Strong induction (5.2) Recursive definitions (5.3) Recurrence Relations (8.1) Principle of Mathematical Induction Want to know

958 views • 21 slides
Natural Deduction and Rule Induction Dr. Liam OConnor University of Edinburgh LFCS UNSW, Term

Natural Deduction and Rule Induction Dr. Liam OConnor University of Edinburgh LFCS UNSW, Term

Natural Deduction Rule Induction Ambiguity Simultaneous Induction Natural Deduction and Rule Induction Dr. Liam OConnor University of Edinburgh LFCS UNSW, Term 3 2020 1 Natural Deduction Rule Induction Ambiguity Simultaneous Induction

614 views • 59 slides
Foundations of Computer Science Lecture 6 Strong Induction Strengthening the Induction

Foundations of Computer Science Lecture 6 Strong Induction Strengthening the Induction

Foundations of Computer Science Lecture 6 Strong Induction Strengthening the Induction Hypothesis Strong Induction Many Flavors of Induction Last Time 1 Proving for all: P ( n ) : 4 n 1 is divisible by 3. n : P ( n )? n i =1

1.51k views • 112 slides
Mathematical Induction 2. Assume the statement is true for any particular value of n and show that

Mathematical Induction 2. Assume the statement is true for any particular value of n and show that

Mathematical Induction http://localhost/~senning/courses/ma229/slides/induction/slide01.html Mathematical Induction http://localhost/~senning/courses/ma229/slides/induction/slide02.html Mathematical Induction prev | slides | next prev | slides

491 views • 3 slides
Foundations of Computer Science Lecture 6 Strong Induction Strengthening the Induction

Foundations of Computer Science Lecture 6 Strong Induction Strengthening the Induction

Foundations of Computer Science Lecture 6 Strong Induction Strengthening the Induction Hypothesis Strong Induction Many Flavors of Induction Last Time 1 Proving for all: P ( n ) : 4 n 1 is divisible by 3. n : P ( n )? n i =1

647 views • 19 slides
Interpreting inductive-inductive definitions as indexed inductive definitions Fredrik Nordvall

Interpreting inductive-inductive definitions as indexed inductive definitions Fredrik Nordvall

Interpreting inductive-inductive definitions as indexed inductive definitions Fredrik Nordvall Forsberg Swansea University csfnf@swansea.ac.uk (Work in progress) What is induction-induction? What is induction-induction? Induction-induction

921 views • 70 slides
Foundations of Computer Science Lecture 5 Induction: Proving For All . . . Induction:

Foundations of Computer Science Lecture 5 Induction: Proving For All . . . Induction:

Foundations of Computer Science Lecture 5 Induction: Proving For All . . . Induction: What and Why? Induction: Good, Bad and Ugly Induction, Well-Ordering and the Smallest Counter-Example Last Time 1 Proving if . . . , then . . .

396 views • 18 slides
Mathematical Induction Examples Strong Induction Induction hypothesis: n k P(n) To prove

Mathematical Induction Examples Strong Induction Induction hypothesis: n k P(n) To prove

Euclid (300 BC) Mathematical Induction Examples Strong Induction Induction hypothesis: n k P(n) To prove n Z + P(n): we prove P(1) (as before) and that k Z + (P(1) P(2) ... P(k)) P(k+1) P(1) P(1) P(2)

343 views • 8 slides
Induction and Its Applications Example for Regular Induction: Correctness of a Decimal-

Induction and Its Applications Example for Regular Induction: Correctness of a Decimal-

10/3/2016 Lecture Outline Review of Induction and Strong Induction Example: A theorem about the first n natural numbers. CSE373: Data Structures and Algorithms Example for Strong Induction: Making Postage Induction and Its

498 views • 4 slides
Associative arrays Associative arrays map a key to a value Keys and values can be different

Associative arrays Associative arrays map a key to a value Keys and values can be different

Associative arrays Associative arrays map a key to a value Keys and values can be different data types name => Emmett arms => 2 42 => The answer Associative arrays can be implemented in many ways

130 views • 8 slides
ECE232: Hardware Organization and Design Lecture 23: Associative Caches Adapted from Computer

ECE232: Hardware Organization and Design Lecture 23: Associative Caches Adapted from Computer

ECE232: Hardware Organization and Design Lecture 23: Associative Caches Adapted from Computer Organization and Design , Patterson & Hennessy, UCB Overview Last time: Direct mapped cache Pretty simple to understand Every memory

339 views • 17 slides
Lecture 1.6: The formal definion of a group Matthew Macauley Department of Mathematical Sciences

Lecture 1.6: The formal definion of a group Matthew Macauley Department of Mathematical Sciences

Lecture 1.6: The formal definion of a group Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 1.6: The formal definition of a

411 views • 7 slides
A unified display proof theory for bunched logic James Brotherston Imperial College London MFPS

A unified display proof theory for bunched logic James Brotherston Imperial College London MFPS

A unified display proof theory for bunched logic James Brotherston Imperial College London MFPS 2010 University of Ottawa, 9 May 2010 Substructural logics: an overview Substructural logics restrict the structural principles of ordinary

673 views • 40 slides
Expressions and Assignment COS 301: Programming Languages UMAINE CIS COS 301 Programming

Expressions and Assignment COS 301: Programming Languages UMAINE CIS COS 301 Programming

Expressions and Assignment COS 301: Programming Languages UMAINE CIS COS 301 Programming Languages Outline Introduction Arithmetic expressions Infix/prefix/postfix Overloaded operators Type conversion Relational

482 views • 16 slides
Cache Performance Associativity Replacement Samira Khan Cache Performance March 28,

Cache Performance Associativity Replacement Samira Khan Cache Performance March 28,

3/28/17 Agenda Review from last lecture Cache access Cache Performance Associativity Replacement Samira Khan Cache Performance March 28, 2017 Direct-Mapped Cache: Placement and Access Cache Abstraction and Metrics 00 | 000

525 views • 16 slides
The Mathematics behind the Property of Associativity An invitation to study the many variants of

The Mathematics behind the Property of Associativity An invitation to study the many variants of

The Mathematics behind the Property of Associativity An invitation to study the many variants of associativity Bruno Teheux and Jean-Luc Marichal University of Luxembourg Associativity for binary functions X , Y non-empty sets F : X X

627 views • 48 slides
IC220: Caching 1 (Chapter 5) 1 Memory, Cost, and Performance Ideal World: we want a memory

IC220: Caching 1 (Chapter 5) 1 Memory, Cost, and Performance Ideal World: we want a memory

IC220: Caching 1 (Chapter 5) 1 Memory, Cost, and Performance Ideal World: we want a memory that is Fast, Big, & Cheap! (Choose any two!) Recent real world situation: SRAM access times are .5 2.5ns at

426 views • 26 slides

More recommend

Explore More Topics

Stay informed with curated content and fresh updates.

animals pets art culture automotive transportation business finance computer internet construction architecture education-career electronics communication

Logo Sambuz

Unleash a World of Digital Possibilities—Browse, Share, and Explore Content Without Boundaries

Useful Links

About Us Privacy Services FAQ's Contact

Newsletter

Stay Informed & Inspired—Subscribe for the Latest Updates, Tips, and Exclusive Content Directly to Your Inbox.

Mail Us

mail@sambuz.com

2026 SAMBUZ, All right reserved.