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