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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:(a−b) = (c−d) 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:

(a−b) = (c−d) 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:(a−b) = (c−d)

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:(a−b) = (c−d) 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.

For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).

For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N.

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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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 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 andtransitive.For reflexivity, note that for all (a,b) ∈ N×N we havea+b = b+a, which means that (a,b)∼ (a,b).For symmetry, let (a,b),(c,d) ∈ N×N. 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 ) ∈ N×N 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:

(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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Constructing the Integers

logo1

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.

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

]

,that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd

+a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′

= a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′

+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′

+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c

=(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′

=(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′

= a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′

= a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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)

]=

[(a′c′+b′d′,a′d′+b′c′)

],

that is, ac+bd +a′d′+b′c′ = a′c′+b′d′+ad +bc.

ac+bd +a′d′+b′c′+b′c=

(a+b′

)c+bd +a′d′+b′c′ =

(a′+b

)c+bd +a′d′+b′c′

= a′c+bc+bd +b′c′+a′d′ = a′(c+d′

)+bc+bd +b′c′

= a′(c′+d

)+bc+bd +b′c′ = a′c′+a′d +bc+bd +b′c′

= 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,[(ac+bd,ad +bc)

]=

[(a′c′+b′d′,a′d′+b′c′)

].

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 N×N 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 N×N 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 N×N 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 N×N 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