Finding the Natural Numbers in the Integers Bernd Schr oder logo1 - PowerPoint PPT Presentation

N ⊆ Z Isomorphism Finding the Natural Numbers in the Integers Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Definition. Let A and B be sets, let n ∈ N , let ◦ 1 ,..., ◦ n be binary operations on A and let ∗ 1 ,..., ∗ n be binary operations on B. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Definition. Let A and B be sets, let n ∈ N , let ◦ 1 ,..., ◦ n be binary operations on A and let ∗ 1 ,..., ∗ n be binary operations on B. Then ( A , ◦ 1 ,..., ◦ n ) is called isomorphic to ( B , ∗ 1 ,..., ∗ n ) iff logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Definition. Let A and B be sets, let n ∈ N , let ◦ 1 ,..., ◦ n be binary operations on A and let ∗ 1 ,..., ∗ n be binary operations on B. Then ( A , ◦ 1 ,..., ◦ n ) is called isomorphic to ( B , ∗ 1 ,..., ∗ n ) iff there is a bijective function f : A → B so that for all k ∈ { 1 ,..., n } and all x , y ∈ A we have that f ( x ◦ k y ) = f ( x ) ∗ k f ( y ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Definition. Let A and B be sets, let n ∈ N , let ◦ 1 ,..., ◦ n be binary operations on A and let ∗ 1 ,..., ∗ n be binary operations on B. Then ( A , ◦ 1 ,..., ◦ n ) is called isomorphic to ( B , ∗ 1 ,..., ∗ n ) iff there is a bijective function f : A → B so that for all k ∈ { 1 ,..., n } and all x , y ∈ A we have that f ( x ◦ k y ) = f ( x ) ∗ k f ( y ) . The function f is called an isomorphism . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ( A , ◦ ) ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ ( A , ◦ ) ( B , ∗ ) ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ ( A , ◦ ) ( B , ∗ ) s ✫ x ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ ( A , ◦ ) ( B , ∗ ) s s y ✫ x ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ ( A , ◦ ) ( B , ∗ ) s s y ✫ x ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ ✻ ( A , ◦ ) ( B , ∗ ) s s y ✫ x ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ s x ◦ y ✻ ( A , ◦ ) ( B , ∗ ) s s y ✫ x ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s q x ◦ y ✻ ( A , ◦ ) ( B , ∗ ) s s y ✫ x ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s s q x ◦ y f ( x ◦ y ) ✻ ( A , ◦ ) ( B , ∗ ) s s y ✫ x ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s s q x ◦ y f ( x ◦ y ) ✻ f ( A , ◦ ) ( B , ∗ ) s q s y ✫ x ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s s q x ◦ y f ( x ◦ y ) ✻ f ( A , ◦ ) ( B , ∗ ) s q s s y ✫ x ✪ ✫ ✪ f ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s s q x ◦ y f ( x ◦ y ) ✻ f ( A , ◦ ) ( B , ∗ ) s s q s s y f ( y ) ✫ x ✪ ✫ ✪ f ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s s q x ◦ y f ( x ◦ y ) ✻ f ( A , ◦ ) ( B , ∗ ) s s q s s y f ( y ) ✫ x ✪ ✫ ✪ f ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s s q x ◦ y f ( x ◦ y ) = f ( x ) ∗ f ( y ) ✻ ✻ f ( A , ◦ ) ( B , ∗ ) s s q s s y f ( y ) ✫ x ✪ ✫ ✪ f ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s s q x ◦ y ✐ f ( x ◦ y ) = f ( x ) ∗ f ( y ) ✻ ✻ f − 1 f ( A , ◦ ) ( B , ∗ ) s s q s s y f ( y ) ✫ x ✪ ✫ ✪ f ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Visualizing Isomorphism ✬ ✩ ✬ ✩ f s s q x ◦ y ✐ f ( x ◦ y ) = f ( x ) ∗ f ( y ) ✻ ✻ f − 1 f ( A , ◦ ) ( B , ∗ ) s s q s s y ✐ f ( y ) ✫ x ✪ ✫ ✪ f − 1 f ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Theorem. The set of natural numbers N , equipped with addition and multiplication, is isomorphic to the subset �� � � ( n + 1 , 1 ) : n ∈ N of the integers Z , equipped with addition and multiplication. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Theorem. The set of natural numbers N , equipped with addition and multiplication, is isomorphic to the subset �� � � ( n + 1 , 1 ) : n ∈ N of the integers Z , equipped with addition �� � � and multiplication. The set ( n + 1 , 1 ) : n ∈ N ⊆ Z will be called N , too logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Theorem. The set of natural numbers N , equipped with addition and multiplication, is isomorphic to the subset �� � � ( n + 1 , 1 ) : n ∈ N of the integers Z , equipped with addition �� � � and multiplication. The set ( n + 1 , 1 ) : n ∈ N ⊆ Z will be called N , too, and we will use the customary digit and place value notation for these numbers. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers

N ⊆ Z Isomorphism Theorem. The set of natural numbers N , equipped with addition and multiplication, is isomorphic to the subset �� � � ( n + 1 , 1 ) : n ∈ N of the integers Z , equipped with addition �� � � and multiplication. The set ( n + 1 , 1 ) : n ∈ N ⊆ Z will be called N , too, and we will use the customary digit and place value notation for these numbers. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finding the Natural Numbers in the Integers