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

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Constructing the Integers

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

What Do We Want?

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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 what they do

Bernd Schr¨

• der

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 what they do (they allow subtraction of arbitrary numbers).

Bernd Schr¨

• der

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 what they do (they allow subtraction of arbitrary numbers).

• 2. Throwing in negative numbers (using “what integers are”)

is harder than it looks.

Bernd Schr¨

• der

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 what they 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.

Bernd Schr¨

• der

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 what they 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.

Bernd Schr¨

• der

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 what they 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.

Bernd Schr¨

• der

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 what they 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.

Bernd Schr¨

• der

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 what they 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:

Bernd Schr¨

• der

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 what they 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)

Bernd Schr¨

• der

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 what they 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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Proposition.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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 and

transitive.

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b).

Bernd Schr¨

• der

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 and

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

Bernd Schr¨

• der

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 and

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

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c, which is equivalent to c+b = d +a

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c, which is equivalent to c+b = d +a, which is equivalent to (c,d) ∼ (a,b).

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c, which is equivalent to c+b = d +a, which is equivalent to (c,d) ∼ (a,b). For transitivity, let (a,b),(c,d),(e,f) ∈ N×N be so that (a,b) ∼ (c,d) and (c,d) ∼ (e,f).

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c, which is equivalent to c+b = d +a, which is equivalent to (c,d) ∼ (a,b). For transitivity, let (a,b),(c,d),(e,f) ∈ N×N be so that (a,b) ∼ (c,d) and (c,d) ∼ (e,f). Then a+d = b+c and c+f = d +e.

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c, which is equivalent to c+b = d +a, which is equivalent to (c,d) ∼ (a,b). For transitivity, let (a,b),(c,d),(e,f) ∈ N×N be so that (a,b) ∼ (c,d) and (c,d) ∼ (e,f). Then a+d = b+c and c+f = d +e. Adding these equations yields a+d +c+f = b+c+d +e.

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c, which is equivalent to c+b = d +a, which is equivalent to (c,d) ∼ (a,b). For transitivity, let (a,b),(c,d),(e,f) ∈ N×N be so that (a,b) ∼ (c,d) and (c,d) ∼ (e,f). Then a+d = b+c and c+f = d +e. Adding these equations yields a+d +c+f = b+c+d +e. We can cancel c+d to obtain a+f = b+e

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c, which is equivalent to c+b = d +a, which is equivalent to (c,d) ∼ (a,b). For transitivity, let (a,b),(c,d),(e,f) ∈ N×N be so that (a,b) ∼ (c,d) and (c,d) ∼ (e,f). Then a+d = b+c and c+f = d +e. Adding these equations yields a+d +c+f = b+c+d +e. We can cancel c+d to obtain a+f = b+e, which means that (a,b) ∼ (e,f).

Bernd Schr¨

• der

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 and

transitive. For reflexivity, note that for all (a,b) ∈ N×N we have a+b = b+a, which means that (a,b) ∼ (a,b). For symmetry, let (a,b),(c,d) ∈ N×N. Then (a,b) ∼ (c,d) is equivalent to a+d = b+c, which is equivalent to c+b = d +a, which is equivalent to (c,d) ∼ (a,b). For transitivity, let (a,b),(c,d),(e,f) ∈ N×N be so that (a,b) ∼ (c,d) and (c,d) ∼ (e,f). Then a+d = b+c and c+f = d +e. Adding these equations yields a+d +c+f = b+c+d +e. We can cancel c+d to obtain a+f = b+e, which means that (a,b) ∼ (e,f).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Motivation for addition:

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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).

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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 ∼.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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:

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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).

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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 ∼.

Bernd Schr¨

• der

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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Proof.

Bernd Schr¨

• der

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′)
• .

Bernd Schr¨

• der

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′)
• Bernd Schr¨
• der

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

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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′

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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 Schr¨

• der

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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Definition.

Bernd Schr¨

• der

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 of

equivalence classes

• (a,b)
• f elements of N×N under the

equivalence relation ∼.

Bernd Schr¨

• der

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 of

equivalence classes

• (a,b)
• f elements of N×N under the

equivalence relation ∼. Addition of integers is defined by

• (a,b)
• +
• (c,d)
• =
• (a+c,b+d)
• Bernd Schr¨
• der

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 of

equivalence classes

• (a,b)
• f 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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Theorem.

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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 +

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Proof (associativity).

Bernd Schr¨

• der

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)
• .

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Proof (neutral element).

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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

Bernd Schr¨

• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Proof (inverse element).

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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)
• =

Bernd Schr¨

• der

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)
• =

=

• (b+a,a+b)
• Bernd Schr¨
• der

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)
• =

=

• (b+a,a+b)
• =
• (b,a)
• +
• (a,b)
• Bernd Schr¨
• der

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)
• =

=

• (b+a,a+b)
• =
• (b,a)
• +
• (a,b)
• =

(−x)+x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

Proof (commutativity).

Bernd Schr¨

• der

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)
• .

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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)
• Bernd Schr¨
• der

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 Schr¨

• der

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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

• Theorem. Multiplication of integers is associative

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

• Theorem. Multiplication of integers is associative, distributive
• ver addition

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

• Theorem. Multiplication of integers is associative, distributive
• ver addition, it has a neutral element 1 :=
• (2,1)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

• Theorem. Multiplication of integers is associative, distributive
• ver addition, it has a neutral element 1 :=
• (2,1)
• , and it is

commutative.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers

logo1 Introduction Equivalence Classes Arithmetic Operations Properties

• Theorem. Multiplication of integers is associative, distributive
• ver addition, it has a neutral element 1 :=
• (2,1)
• , and it is

commutative.

• Proof. Exercise.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Constructing the Integers