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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

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