Constructing the Integers Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c , which is equivalent to c + b = d + a logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c , which is equivalent to c + b = d + a , which is equivalent to ( c , d ) ∼ ( a , b ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c , which is equivalent to c + b = d + a , which is equivalent to ( c , d ) ∼ ( a , b ) . For transitivity, let ( a , b ) , ( c , d ) , ( e , f ) ∈ N × N be so that ( a , b ) ∼ ( c , d ) and ( c , d ) ∼ ( e , f ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c , which is equivalent to c + b = d + a , which is equivalent to ( c , d ) ∼ ( a , b ) . For transitivity, let ( a , b ) , ( c , d ) , ( e , f ) ∈ N × N be so that ( a , b ) ∼ ( c , d ) and ( c , d ) ∼ ( e , f ) . Then a + d = b + c and c + f = d + e . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c , which is equivalent to c + b = d + a , which is equivalent to ( c , d ) ∼ ( a , b ) . For transitivity, let ( a , b ) , ( c , d ) , ( e , f ) ∈ N × N be so that ( a , b ) ∼ ( c , d ) and ( c , d ) ∼ ( e , f ) . Then a + d = b + c and c + f = d + e . Adding these equations yields a + d + c + f = b + c + d + e . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c , which is equivalent to c + b = d + a , which is equivalent to ( c , d ) ∼ ( a , b ) . For transitivity, let ( a , b ) , ( c , d ) , ( e , f ) ∈ N × N be so that ( a , b ) ∼ ( c , d ) and ( c , d ) ∼ ( e , f ) . Then a + d = b + c and c + f = d + e . Adding these equations yields a + d + c + f = b + c + d + e . We can cancel c + d to obtain a + f = b + e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c , which is equivalent to c + b = d + a , which is equivalent to ( c , d ) ∼ ( a , b ) . For transitivity, let ( a , b ) , ( c , d ) , ( e , f ) ∈ N × N be so that ( a , b ) ∼ ( c , d ) and ( c , d ) ∼ ( e , f ) . Then a + d = b + c and c + f = d + e . Adding these equations yields a + d + c + f = b + c + d + e . We can cancel c + d to obtain a + f = b + e , which means that ( a , b ) ∼ ( e , f ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proposition. The relation ∼ on N × N defined by ( a , b ) ∼ ( c , d ) iff a + d = b + c is an equivalence relation. Proof. We must prove that ∼ is reflexive, symmetric and transitive. For reflexivity, note that for all ( a , b ) ∈ N × N we have a + b = b + a , which means that ( a , b ) ∼ ( a , b ) . For symmetry, let ( a , b ) , ( c , d ) ∈ N × N . Then ( a , b ) ∼ ( c , d ) is equivalent to a + d = b + c , which is equivalent to c + b = d + a , which is equivalent to ( c , d ) ∼ ( a , b ) . For transitivity, let ( a , b ) , ( c , d ) , ( e , f ) ∈ N × N be so that ( a , b ) ∼ ( c , d ) and ( c , d ) ∼ ( e , f ) . Then a + d = b + c and c + f = d + e . Adding these equations yields a + d + c + f = b + c + d + e . We can cancel c + d to obtain a + f = b + e , which means that ( a , b ) ∼ ( e , f ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) = ac logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) = ac − ad logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) = ac − ad − bc logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) = ac − ad − bc + bd logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) = ac − ad − bc + bd = ( ac + bd ) − ( ad + bc ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) = ac − ad − bc + bd = ( ac + bd ) − ( ad + bc ) . Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) = ac − ad − bc + bd = ( ac + bd ) − ( ad + bc ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Motivation for addition: ( a − b )+( c − d ) = ( a + c ) − ( b + d ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) + ( c , d ) : = ( a + c , b + d ) is well-defined. Proof. Exercise. Motivation for multiplication: ( a − b ) · ( c − d ) = ac − ad − bc + bd = ( ac + bd ) − ( ad + bc ) . � � Proposition. For each ( x , y ) ∈ N × N , let ( x , y ) denote the equivalence class of ( x , y ) under ∼ . Then the operation � � � � � � ( a , b ) ( c , d ) : = ( ac + bd , ad + bc ) is well-defined. · logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d a ′ � � + bc + bd + b ′ c ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b � � d + bc + b ′ c ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ d + bc + b ′ c ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ � d + c ′ � = + bc logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ � d + c ′ � = + bc a ′ c ′ + ad + b ′ � d ′ + c � = + bc logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ � d + c ′ � = + bc a ′ c ′ + ad + b ′ � d ′ + c + bc = a ′ c ′ + ad + b ′ d ′ + b ′ c + bc � = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ � d + c ′ � = + bc a ′ c ′ + ad + b ′ � d ′ + c + bc = a ′ c ′ + ad + b ′ d ′ + b ′ c + bc � = a ′ c ′ + b ′ d ′ + ad + bc + b ′ c = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ � d + c ′ � = + bc a ′ c ′ + ad + b ′ � d ′ + c + bc = a ′ c ′ + ad + b ′ d ′ + b ′ c + bc � = a ′ c ′ + b ′ d ′ + ad + bc + b ′ c = Hence ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ � d + c ′ � = + bc a ′ c ′ + ad + b ′ � d ′ + c + bc = a ′ c ′ + ad + b ′ d ′ + b ′ c + bc � = a ′ c ′ + b ′ d ′ + ad + bc + b ′ c = Hence ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc , that is, ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � ( ac + bd , ad + bc ) = . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � � � � � � � ( a , b ) = ( a ′ , b ′ ) ( c , d ) = ( c ′ , d ′ ) Proof. Let and let . We ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � must prove ( ac + bd , ad + bc ) = , that is, ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc . ac + bd + a ′ d ′ + b ′ c ′ + b ′ c c + bd + a ′ d ′ + b ′ c ′ = a ′ + b c + bd + a ′ d ′ + b ′ c ′ � a + b ′ � � � = a ′ c + bc + bd + b ′ c ′ + a ′ d ′ = a ′ � c + d ′ � + bc + bd + b ′ c ′ = c ′ + d + bc + bd + b ′ c ′ = a ′ c ′ + a ′ d + bc + bd + b ′ c ′ a ′ � � = a ′ c ′ + a ′ + b d + bc + b ′ c ′ = a ′ c ′ + � � � a + b ′ � d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ d + bc + b ′ c ′ = a ′ c ′ + ad + b ′ � d + c ′ � = + bc a ′ c ′ + ad + b ′ � d ′ + c + bc = a ′ c ′ + ad + b ′ d ′ + b ′ c + bc � = a ′ c ′ + b ′ d ′ + ad + bc + b ′ c = Hence ac + bd + a ′ d ′ + b ′ c ′ = a ′ c ′ + b ′ d ′ + ad + bc , that is, ( a ′ c ′ + b ′ d ′ , a ′ d ′ + b ′ c ′ ) � � � � ( ac + bd , ad + bc ) = . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Definition. The integers Z are defined to be the set of � � equivalence classes ( a , b ) of elements of N × N under the equivalence relation ∼ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Definition. The integers Z are defined to be the set of � � equivalence classes ( a , b ) of elements of N × N under the equivalence relation ∼ . Addition of integers is defined by � � � � � � ( a , b ) + ( c , d ) = ( a + c , b + d ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Definition. The integers Z are defined to be the set of � � equivalence classes ( a , b ) of elements of N × N under the equivalence relation ∼ . Addition of integers is defined by � � � � � � ( a , b ) + ( c , d ) = ( a + c , b + d ) and multiplication is � � � � � � ( a , b ) ( c , d ) = ( ac + bd , ad + bc ) defined by . · logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Theorem. The addition + of integers is associative logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Theorem. The addition + of integers is associative, � � 0 : = ( 1 , 1 ) is a neutral element with respect to + logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Theorem. The addition + of integers is associative, � � 0 : = ( 1 , 1 ) is a neutral element with respect to + , for every � � � � x = ( a , b ) ∈ Z there is an element − x : = ( b , a ) so that x +( − x ) = ( − x )+ x = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Theorem. The addition + of integers is associative, � � 0 : = ( 1 , 1 ) is a neutral element with respect to + , for every � � � � x = ( a , b ) ∈ Z there is an element − x : = ( b , a ) so that x +( − x ) = ( − x )+ x = 0 , and + is commutative. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proof (associativity). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . Then ( x + y )+ z logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . Then �� �� � � � � ( x + y )+ z = ( a , b ) + ( c , d ) + ( e , f ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . Then �� �� � � � � ( x + y )+ z = ( a , b ) + ( c , d ) + ( e , f ) � � � � = ( a + c , b + d ) + ( e , f ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . Then �� �� � � � � ( x + y )+ z = ( a , b ) + ( c , d ) + ( e , f ) � � � � = ( a + c , b + d ) + ( e , f ) �� �� = ( a + c )+ e , ( b + d )+ f logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . Then �� �� � � � � ( x + y )+ z = ( a , b ) + ( c , d ) + ( e , f ) � � � � = ( a + c , b + d ) + ( e , f ) �� �� = ( a + c )+ e , ( b + d )+ f �� �� = a +( c + e ) , b +( d + f ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . Then �� �� � � � � ( x + y )+ z = ( a , b ) + ( c , d ) + ( e , f ) � � � � = ( a + c , b + d ) + ( e , f ) �� �� = ( a + c )+ e , ( b + d )+ f �� �� = a +( c + e ) , b +( d + f ) � � � � = ( a , b ) + ( c + e , d + f ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . Then �� �� � � � � ( x + y )+ z = ( a , b ) + ( c , d ) + ( e , f ) � � � � = ( a + c , b + d ) + ( e , f ) �� �� = ( a + c )+ e , ( b + d )+ f �� �� = a +( c + e ) , b +( d + f ) � � � � = ( a , b ) + ( c + e , d + f ) �� �� � � � � = ( a , b ) + ( c , d ) + ( e , f ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (associativity). Let x , y , z ∈ Z with x = ( a , b ) , � � � � y = ( c , d ) , and z = ( e , f ) . Then �� �� � � � � ( x + y )+ z = ( a , b ) + ( c , d ) + ( e , f ) � � � � = ( a + c , b + d ) + ( e , f ) �� �� = ( a + c )+ e , ( b + d )+ f �� �� = a +( c + e ) , b +( d + f ) � � � � = ( a , b ) + ( c + e , d + f ) �� �� � � � � = ( a , b ) + ( c , d ) + ( e , f ) = x +( y + z ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties Proof (neutral element). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (neutral element). Let x = ( a , b ) ∈ Z . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (neutral element). Let x = ( a , b ) ∈ Z . x + 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (neutral element). Let x = ( a , b ) ∈ Z . � � � � x + 0 = ( a , b ) + ( 1 , 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (neutral element). Let x = ( a , b ) ∈ Z . � � � � x + 0 = ( a , b ) + ( 1 , 1 ) � � = ( a + 1 , b + 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (neutral element). Let x = ( a , b ) ∈ Z . � � � � x + 0 = ( a , b ) + ( 1 , 1 ) � � = ( a + 1 , b + 1 ) � � = ( a , b ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (neutral element). Let x = ( a , b ) ∈ Z . � � � � x + 0 = ( a , b ) + ( 1 , 1 ) � � = ( a + 1 , b + 1 ) � � = ( a , b ) = x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (neutral element). Let x = ( a , b ) ∈ Z . � � � � x + 0 = ( a , b ) + ( 1 , 1 ) � � = ( a + 1 , b + 1 ) � � = ( a , b ) = x � � = ( a , b ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers

Introduction Equivalence Classes Arithmetic Operations Properties � � Proof (neutral element). Let x = ( a , b ) ∈ Z . � � � � x + 0 = ( a , b ) + ( 1 , 1 ) � � = ( a + 1 , b + 1 ) � � = ( a , b ) = x � � = ( a , b ) � � = ( 1 + a , 1 + b ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Constructing the Integers