int_const
DESCRIPTION
construccionTRANSCRIPT
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Constructing the Integers
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?
1. We know what the integers are (natural numbers,negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are
(natural numbers,negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero)
and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do
(they allow subtraction of arbitrary numbers).2. Throwing in negative numbers (using what integers are)
is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.
2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.
2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.
2.3 Some elementary and middle school students struggle withthe concept.
3. So we will focus on what the integers do, that is, we willfocus on formal differences.
Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.
3. So we will focus on what the integers do, that is, we willfocus on formal differences.
Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.
Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:
(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd)
iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
What Do We Want?1. We know what the integers are (natural numbers,
negative natural numbers and zero) and we know whatthey do (they allow subtraction of arbitrary numbers).
2. Throwing in negative numbers (using what integers are)is harder than it looks.2.1 Construction has a lot of case distinctions.2.2 Ancient Greek philosophers avoided negative quantities.2.3 Some elementary and middle school students struggle with
the concept.3. So we will focus on what the integers do, that is, we will
focus on formal differences.Motivation for the formal definition of the integers:(ab) = (cd) iff a+d = b+ c.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition.
The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof.
We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.
For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).
For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN.
Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d)
isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ).
Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e.
Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e.
We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e
, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proposition. The relation on NN defined by (a,b) (c,d)iff a+d = b+ c is an equivalence relation.
Proof. We must prove that is reflexive, symmetric andtransitive.For reflexivity, note that for all (a,b) NN we havea+b = b+a, which means that (a,b) (a,b).For symmetry, let (a,b),(c,d) NN. Then (a,b) (c,d) isequivalent 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 ) NN be so that(a,b) (c,d) and (c,d) (e, f ). Then a+d = b+ c andc+ f = d+ e. Adding these equations yieldsa+d+ c+ f = b+ c+d+ e. We can cancel c+d to obtaina+ f = b+ e, which means that (a,b) (e, f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition:
(ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd)
= (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition.
For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:
(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd)
= acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = ac
adbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acad
bc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc
+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd
= (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition.
For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Motivation for addition: (ab)+(cd) = (a+ c) (b+d).
Proposition. For each (x,y) NN, 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:(ab) (cd) = acadbc+bd = (ac+bd) (ad+bc).
Proposition. For each (x,y) NN, 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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof.
Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
].
Wemust prove
[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
]
,that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd
+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc
= ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd
+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc
+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc
=(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc
=(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad
= a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc
= ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc
= ac+(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc
= ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc
= ac+ad+bd+bc+bc= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc
, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof. Let[(a,b)
]=
[(a,b)
]and let
[(c,d)
]=
[(c,d)
]. We
must prove[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
],
that is, ac+bd+ad+bc = ac+bd+ad+bc.
ac+bd+ad+bc+bc=
(a+b
)c+bd+ad+bc =
(a+b
)c+bd+ad+bc
= ac+bc+bd+bc+ad = a(c+d
)+bc+bd+bc
= a(c+d
)+bc+bd+bc = ac+ad+bc+bd+bc
= ac+(a+b
)d+bc+bc = ac+
(a+b
)d+bc+bc
= ac+ad+bd+bc+bc = ac+ad+b(d+ c
)+bc
= ac+ad+b(d+ c
)+bc = ac+ad+bd+bc+bc
= ac+bd+ad+bc+bc
Hence ac+bd+ad+bc = ac+bd+ad+bc, that is,[(ac+bd,ad+bc)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Definition.
The integers Z are defined to be the set ofequivalence classes
[(a,b)
]of elements of NN under the
equivalence relation . Addition of integers is defined by[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]and multiplication is
defined by[(a,b)
][(c,d)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Definition. The integers Z are defined to be the set ofequivalence classes
[(a,b)
]of elements of NN under the
equivalence relation .
Addition of integers is defined by[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]and multiplication is
defined by[(a,b)
][(c,d)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Definition. The integers Z are defined to be the set ofequivalence classes
[(a,b)
]of elements of NN under the
equivalence relation . Addition of integers is defined by[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]
and multiplication isdefined by
[(a,b)
][(c,d)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Definition. The integers Z are defined to be the set ofequivalence classes
[(a,b)
]of elements of NN under the
equivalence relation . Addition of integers is defined by[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]and multiplication is
defined by[(a,b)
][(c,d)
]=
[(ac+bd,ad+bc)
].
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Theorem. The addition + of integers is associative,0 :=
[(1,1)
]is a neutral element with respect to +
, for everyx =
[(a,b)
] Z there is an element x :=
[(b,a)
]so that
x+(x) = (x)+ x = 0, and + is commutative.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]
=[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]
= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
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)
]=
[(1,1)
]+[(a,b)
]= 0+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element).
Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x) =
[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]=
[(1,1)
]= 0=
[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.
x+(x) =[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]=
[(1,1)
]= 0=
[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x)
=[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]=
[(1,1)
]= 0=
[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x) =
[(a,b)
]+[(b,a)
]
=[(a+b,b+a)
]=
[(1,1)
]= 0=
[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x) =
[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]
=[(1,1)
]= 0=
[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x) =
[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]=
[(1,1)
]
= 0=
[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x) =
[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]=
[(1,1)
]= 0
=[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x) =
[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]=
[(1,1)
]= 0=
[(b+a,a+b)
]
=[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x) =
[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]=
[(1,1)
]= 0=
[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]
= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (inverse element). Let x =[(a,b)
] Z and let
x :=[(b,a)
] Z.x+(x) =
[(a,b)
]+[(b,a)
]=
[(a+b,b+a)
]=
[(1,1)
]= 0=
[(b+a,a+b)
]=
[(b,a)
]+[(a,b)
]= (x)+ x.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity).
Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y =[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]=
[(c+a,d+b)
]=
[(c,d)
]+[(a,b)
]= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity). Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y =[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]=
[(c+a,d+b)
]=
[(c,d)
]+[(a,b)
]= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity). Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y
=[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]=
[(c+a,d+b)
]=
[(c,d)
]+[(a,b)
]= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity). Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y =[(a,b)
]+[(c,d)
]
=[(a+ c,b+d)
]=
[(c+a,d+b)
]=
[(c,d)
]+[(a,b)
]= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity). Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y =[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]
=[(c+a,d+b)
]=
[(c,d)
]+[(a,b)
]= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity). Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y =[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]=
[(c+a,d+b)
]
=[(c,d)
]+[(a,b)
]= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity). Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y =[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]=
[(c+a,d+b)
]=
[(c,d)
]+[(a,b)
]
= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity). Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y =[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]=
[(c+a,d+b)
]=
[(c,d)
]+[(a,b)
]= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Proof (commutativity). Let x,y,z Z with x =[(a,b)
]and
y =[(c,d)
].
x+ y =[(a,b)
]+[(c,d)
]=
[(a+ c,b+d)
]=
[(c+a,d+b)
]=
[(c,d)
]+[(a,b)
]= y+ x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Theorem. Multiplication of integers is associative
, distributiveover addition, it has a neutral element 1 :=
[(2,1)
], and it is
commutative.
Proof. Exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Theorem. Multiplication of integers is associative, distributiveover addition
, it has a neutral element 1 :=[(2,1)
], and it is
commutative.
Proof. Exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Theorem. Multiplication of integers is associative, distributiveover addition, it has a neutral element 1 :=
[(2,1)
]
, and it iscommutative.
Proof. Exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Theorem. Multiplication of integers is associative, distributiveover addition, it has a neutral element 1 :=
[(2,1)
], and it is
commutative.
Proof. Exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
-
logo1
Introduction Equivalence Classes Arithmetic Operations Properties
Theorem. Multiplication of integers is associative, distributiveover addition, it has a neutral element 1 :=
[(2,1)
], and it is
commutative.
Proof. Exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constructing the Integers
IntroductionEquivalence ClassesArithmetic OperationsProperties