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Finite vs. Infinite Countability Examples Countable Sets Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Countable Sets Finite vs. Infinite Countability Examples Introduction

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SLIDE 1

logo1 Finite vs. Infinite Countability Examples

Countable Sets

Bernd Schr¨

  • der

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 2

logo1 Finite vs. Infinite Countability Examples

Introduction

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 3

logo1 Finite vs. Infinite Countability Examples

Introduction

  • 1. Sets are equivalent iff “they are of the same size”

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 4

logo1 Finite vs. Infinite Countability Examples

Introduction

  • 1. Sets are equivalent iff “they are of the same size” (more

precisely: iff there is a bijective function between them).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 5

logo1 Finite vs. Infinite Countability Examples

Introduction

  • 1. Sets are equivalent iff “they are of the same size” (more

precisely: iff there is a bijective function between them).

  • 2. A set F is called finite iff F is empty or there is an n ∈ N

and a bijective function f : {1,...,n} → F.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 6

logo1 Finite vs. Infinite Countability Examples

Introduction

  • 1. Sets are equivalent iff “they are of the same size” (more

precisely: iff there is a bijective function between them).

  • 2. A set F is called finite iff F is empty or there is an n ∈ N

and a bijective function f : {1,...,n} → F.

  • 3. For finite sets F = /

0 we set |F| := n with n as above and we set |/ 0| := 0.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 7

logo1 Finite vs. Infinite Countability Examples

Introduction

  • 1. Sets are equivalent iff “they are of the same size” (more

precisely: iff there is a bijective function between them).

  • 2. A set F is called finite iff F is empty or there is an n ∈ N

and a bijective function f : {1,...,n} → F.

  • 3. For finite sets F = /

0 we set |F| := n with n as above and we set |/ 0| := 0.

  • 4. Sets that are not finite are called infinite.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 8

logo1 Finite vs. Infinite Countability Examples

Introduction

  • 1. Sets are equivalent iff “they are of the same size” (more

precisely: iff there is a bijective function between them).

  • 2. A set F is called finite iff F is empty or there is an n ∈ N

and a bijective function f : {1,...,n} → F.

  • 3. For finite sets F = /

0 we set |F| := n with n as above and we set |/ 0| := 0.

  • 4. Sets that are not finite are called infinite.
  • 5. For infinite sets I we set |I| := ∞.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 9

logo1 Finite vs. Infinite Countability Examples

Introduction

  • 1. Sets are equivalent iff “they are of the same size” (more

precisely: iff there is a bijective function between them).

  • 2. A set F is called finite iff F is empty or there is an n ∈ N

and a bijective function f : {1,...,n} → F.

  • 3. For finite sets F = /

0 we set |F| := n with n as above and we set |/ 0| := 0.

  • 4. Sets that are not finite are called infinite.
  • 5. For infinite sets I we set |I| := ∞.
  • 6. If A ⊆ B, A,B are finite and |A| = |B|, then A = B.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 10

logo1 Finite vs. Infinite Countability Examples

Introduction

  • 1. Sets are equivalent iff “they are of the same size” (more

precisely: iff there is a bijective function between them).

  • 2. A set F is called finite iff F is empty or there is an n ∈ N

and a bijective function f : {1,...,n} → F.

  • 3. For finite sets F = /

0 we set |F| := n with n as above and we set |/ 0| := 0.

  • 4. Sets that are not finite are called infinite.
  • 5. For infinite sets I we set |I| := ∞.
  • 6. If A ⊆ B, A,B are finite and |A| = |B|, then A = B.

For infinite sets, the situation is different.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 11

logo1 Finite vs. Infinite Countability Examples

Theorem.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 12

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 13

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

1 2 3

. . .

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 14

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 15

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

✲

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 16

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

✲ ♦

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 17

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

✲ ✲ ♦

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 18

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

✲ ✲ ♦ ♦

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 19

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

✲ ✲ ✲ ♦ ♦

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 20

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

✲ ✲ ✲ ♦ ♦ ♦

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 21

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

✲ ✲ ✲ ✲ ♦ ♦ ♦

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 22

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

−1 −2 −3 −4 1 2 3

. . . . . .

✲ ✲ ✲ ✲ ♦ ♦ ♦ ♦

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 23

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 24

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.

Proof.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 25

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.
  • Proof. The function f(n) :=
  • n−1

2 ;

if n is odd, −n

2;

if n is even,

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 26

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.
  • Proof. The function f(n) :=
  • n−1

2 ;

if n is odd, −n

2;

if n is even, is bijective.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 27

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.
  • Proof. The function f(n) :=
  • n−1

2 ;

if n is odd, −n

2;

if n is even, is bijective. (Good exercise.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 28

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.
  • Proof. The function f(n) :=
  • n−1

2 ;

if n is odd, −n

2;

if n is even, is bijective. (Good exercise.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 29

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.
  • Proof. The function f(n) :=
  • n−1

2 ;

if n is odd, −n

2;

if n is even, is bijective. (Good exercise.) Definition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 30

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.
  • Proof. The function f(n) :=
  • n−1

2 ;

if n is odd, −n

2;

if n is even, is bijective. (Good exercise.)

  • Definition. A set C is called countably infinite iff there is a

bijective function f : N → C.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 31

logo1 Finite vs. Infinite Countability Examples

  • Theorem. N is equivalent to Z.
  • Proof. The function f(n) :=
  • n−1

2 ;

if n is odd, −n

2;

if n is even, is bijective. (Good exercise.)

  • Definition. A set C is called countably infinite iff there is a

bijective function f : N → C. A set C is called countable iff C is finite or countably infinite.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 32

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 33

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.

Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 34

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 35

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S].

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 36

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 37

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 38

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 39

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 40

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective (and it really maps into S).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 41

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective (and it really maps into S). Suppose for a contradiction that g is not surjective.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-42
SLIDE 42

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective (and it really maps into S). Suppose for a contradiction that g is not surjective. Let b := minf −1 S\g[N]

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-43
SLIDE 43

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective (and it really maps into S). Suppose for a contradiction that g is not surjective. Let b := minf −1 S\g[N]

  • and let k :=
  • f −1[S]∩{1,...,b−1}
  • .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 44

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective (and it really maps into S). Suppose for a contradiction that g is not surjective. Let b := minf −1 S\g[N]

  • and let k :=
  • f −1[S]∩{1,...,b−1}
  • .

Then f −1[S]∩{1,...,b−1} = {n1,...,nk}

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 45

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective (and it really maps into S). Suppose for a contradiction that g is not surjective. Let b := minf −1 S\g[N]

  • and let k :=
  • f −1[S]∩{1,...,b−1}
  • .

Then f −1[S]∩{1,...,b−1} = {n1,...,nk} and b = min

  • f −1[S]\{n1,...,nk}
  • .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-46
SLIDE 46

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective (and it really maps into S). Suppose for a contradiction that g is not surjective. Let b := minf −1 S\g[N]

  • and let k :=
  • f −1[S]∩{1,...,b−1}
  • .

Then f −1[S]∩{1,...,b−1} = {n1,...,nk} and b = min

  • f −1[S]\{n1,...,nk}
  • . But then b = nk+1,

contradiction.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-47
SLIDE 47

logo1 Finite vs. Infinite Countability Examples

  • Theorem. If C is countable and S ⊆ C, then S is countable.
  • Proof. WLOG let S be infinite. Let f : N → C be bijective and

let n1 := minf −1[S]. For k ∈ N, define nk+1 recursively. Once n1,...,nk are chosen, let nk+1:=min

  • f −1[S]\{n1,...,nk}
  • .

Define g : N → S by g(k) := f(nk). Then g is injective (and it really maps into S). Suppose for a contradiction that g is not surjective. Let b := minf −1 S\g[N]

  • and let k :=
  • f −1[S]∩{1,...,b−1}
  • .

Then f −1[S]∩{1,...,b−1} = {n1,...,nk} and b = min

  • f −1[S]\{n1,...,nk}
  • . But then b = nk+1,

contradiction.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-48
SLIDE 48

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-49
SLIDE 49

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-50
SLIDE 50

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 2 3 4 m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-51
SLIDE 51

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4 m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-52
SLIDE 52

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4 r r r r r r r r r r r r r r r r m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-53
SLIDE 53

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4

(1,2) (2,2) (3,2) (4,2) (1,3) (1,4) (2,3) (2,4) (3,3) (3,4) (4,3) (4,4) (1,1) (2,1) (3,1) (4,1)

r r r r r r r r r r r r r r r r m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-54
SLIDE 54

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4

··· . . .

(1,2) (2,2) (3,2) (4,2) (1,3) (1,4) (2,3) (2,4) (3,3) (3,4) (4,3) (4,4) (1,1) (2,1) (3,1) (4,1)

···

r r r r r r r r r r r r r r r r m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-55
SLIDE 55

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4

··· . . .

first one (1,2) (2,2) (3,2) (4,2) (1,3) (1,4) (2,3) (2,4) (3,3) (3,4) (4,3) (4,4) (1,1) (2,1) (3,1) (4,1)

···

■ r r r r r r r r r r r r r r r r m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-56
SLIDE 56

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4

··· . . .

first one next two (1,2) (2,2) (3,2) (4,2) (1,3) (1,4) (2,3) (2,4) (3,3) (3,4) (4,3) (4,4) (1,1) (2,1) (3,1) (4,1)

···

■ ■ r r r r r r r r r r r r r r r r m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-57
SLIDE 57

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4

··· . . .

first one next two next three (1,2) (2,2) (3,2) (4,2) (1,3) (1,4) (2,3) (2,4) (3,3) (3,4) (4,3) (4,4) (1,1) (2,1) (3,1) (4,1)

···

■ ■ ■ r r r r r r r r r r r r r r r r m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-58
SLIDE 58

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4

··· . . .

first one next two next three next four (1,2) (2,2) (3,2) (4,2) (1,3) (1,4) (2,3) (2,4) (3,3) (3,4) (4,3) (4,4) (1,1) (2,1) (3,1) (4,1)

···

■ ■ ■ ■ r r r r r r r r r r r r r r r r m n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-59
SLIDE 59

logo1 Finite vs. Infinite Countability Examples

  • Theorem. The set N×N is countable

✻ ✲ 1 1 2 3 4 2 3 4

··· . . .

first one next two next three next four (1,2) (2,2) (3,2) (4,2) (1,3) (1,4) (2,3) (2,4) (3,3) (3,4) (4,3) (4,4) (1,1) (2,1) (3,1) (4,1)

···

■ ■ ■ ■ r r r r r r r r r r r r r r r r m n f(m,n) := 1 2(m+n−1)(m+n−2)+n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 60

logo1 Finite vs. Infinite Countability Examples

Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 61

logo1 Finite vs. Infinite Countability Examples

  • Proof. We claim that the function f : N×N → N defined by

f(m,n) := 1 2(m+n−1)(m+n−2)+n is bijective.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-62
SLIDE 62

logo1 Finite vs. Infinite Countability Examples

  • Proof. We claim that the function f : N×N → N defined by

f(m,n) := 1 2(m+n−1)(m+n−2)+n is bijective. First, an auxiliary equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 63

logo1 Finite vs. Infinite Countability Examples

  • Proof. We claim that the function f : N×N → N defined by

f(m,n) := 1 2(m+n−1)(m+n−2)+n is bijective. First, an auxiliary equation. f(m−1,n+1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-64
SLIDE 64

logo1 Finite vs. Infinite Countability Examples

  • Proof. We claim that the function f : N×N → N defined by

f(m,n) := 1 2(m+n−1)(m+n−2)+n is bijective. First, an auxiliary equation. f(m−1,n+1) = 1 2

  • (m−1)+(n+1)−1
  • (m−1)+(n+1)−2
  • +n+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-65
SLIDE 65

logo1 Finite vs. Infinite Countability Examples

  • Proof. We claim that the function f : N×N → N defined by

f(m,n) := 1 2(m+n−1)(m+n−2)+n is bijective. First, an auxiliary equation. f(m−1,n+1) = 1 2

  • (m−1)+(n+1)−1
  • (m−1)+(n+1)−2
  • +n+1

= 1 2(m+n−1)(m+n−2)+n+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-66
SLIDE 66

logo1 Finite vs. Infinite Countability Examples

  • Proof. We claim that the function f : N×N → N defined by

f(m,n) := 1 2(m+n−1)(m+n−2)+n is bijective. First, an auxiliary equation. f(m−1,n+1) = 1 2

  • (m−1)+(n+1)−1
  • (m−1)+(n+1)−2
  • +n+1

= 1 2(m+n−1)(m+n−2)+n+1 = f(m,n)+1.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-67
SLIDE 67

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-68
SLIDE 68

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity). We will prove by induction on m that f(m,n) = f(a,b) implies (m,n) = (a,b).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-69
SLIDE 69

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity). We will prove by induction on m that f(m,n) = f(a,b) implies (m,n) = (a,b). Base step, m = 1:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-70
SLIDE 70

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity). We will prove by induction on m that f(m,n) = f(a,b) implies (m,n) = (a,b). Base step, m = 1: Let f(1,n) = f(a,b).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-71
SLIDE 71

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity). We will prove by induction on m that f(m,n) = f(a,b) implies (m,n) = (a,b). Base step, m = 1: Let f(1,n) = f(a,b). Then f(1,n) = f(a,b)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-72
SLIDE 72

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity). We will prove by induction on m that f(m,n) = f(a,b) implies (m,n) = (a,b). Base step, m = 1: Let f(1,n) = f(a,b). Then f(1,n) = f(a,b) 1 2(1+n−1)(1+n−2)+n = 1 2(a+b−1)(a+b−2)+b

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-73
SLIDE 73

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity). We will prove by induction on m that f(m,n) = f(a,b) implies (m,n) = (a,b). Base step, m = 1: Let f(1,n) = f(a,b). Then f(1,n) = f(a,b) 1 2(1+n−1)(1+n−2)+n = 1 2(a+b−1)(a+b−2)+b n(n−1)+2n = (a+b−1)(a+b−2)+2b

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-74
SLIDE 74

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity). We will prove by induction on m that f(m,n) = f(a,b) implies (m,n) = (a,b). Base step, m = 1: Let f(1,n) = f(a,b). Then f(1,n) = f(a,b) 1 2(1+n−1)(1+n−2)+n = 1 2(a+b−1)(a+b−2)+b n(n−1)+2n = (a+b−1)(a+b−2)+2b n2 +n = (a+b−1)2 −(a+b−1)+2b

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-75
SLIDE 75

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity). We will prove by induction on m that f(m,n) = f(a,b) implies (m,n) = (a,b). Base step, m = 1: Let f(1,n) = f(a,b). Then f(1,n) = f(a,b) 1 2(1+n−1)(1+n−2)+n = 1 2(a+b−1)(a+b−2)+b n(n−1)+2n = (a+b−1)(a+b−2)+2b n2 +n = (a+b−1)2 −(a+b−1)+2b n2 +n = (a+b−1)2 +(b−a+1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-76
SLIDE 76

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 77

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-78
SLIDE 78

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-79
SLIDE 79

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-80
SLIDE 80

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n, contradiction.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-81
SLIDE 81

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-82
SLIDE 82

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-83
SLIDE 83

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-84
SLIDE 84

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-85
SLIDE 85

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-86
SLIDE 86

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-87
SLIDE 87

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-88
SLIDE 88

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-89
SLIDE 89

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1, which implies that 2b = (−2k +2)n+k −k2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-90
SLIDE 90

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1, which implies that 2b = (−2k +2)n+k −k2 ≤ 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-91
SLIDE 91

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1, which implies that 2b = (−2k +2)n+k −k2 ≤ 0, contradiction.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-92
SLIDE 92

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1, which implies that 2b = (−2k +2)n+k −k2 ≤ 0, contradiction. Thus a+b−1 = n.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-93
SLIDE 93

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1, which implies that 2b = (−2k +2)n+k −k2 ≤ 0, contradiction. Thus a+b−1 = n. Hence b−a+1 = n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-94
SLIDE 94

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1, which implies that 2b = (−2k +2)n+k −k2 ≤ 0, contradiction. Thus a+b−1 = n. Hence b−a+1 = n, and then 2b = 2n and a = 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-95
SLIDE 95

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1, which implies that 2b = (−2k +2)n+k −k2 ≤ 0, contradiction. Thus a+b−1 = n. Hence b−a+1 = n, and then 2b = 2n and a = 1, as was to be proved.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-96
SLIDE 96

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, cont.). n2 +n = (a+b−1)2 +(b−a+1) Suppose for a contradiction that a+b−1 < n. Then b−a+1 ≤ b < n and (a+b−1)2 +(b−a+1) < n2 +n,

  • contradiction. Thus a+b−1 ≥ n.

Suppose for a contradiction that a+b−1 = n+k for some k ∈ N. Then (a+b−1)2 = (n+k)2 = n2 +2kn+k2, so that b−a+1 = −(2k −1)n−k2. Hence n+k −b+1 = a = (2k −1)n+k2 +b+1, which implies that 2b = (−2k +2)n+k −k2 ≤ 0, contradiction. Thus a+b−1 = n. Hence b−a+1 = n, and then 2b = 2n and a = 1, as was to be proved. Hence f(1,n) = f(a,b) implies (1,n) = (a,b).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-97
SLIDE 97

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-98
SLIDE 98

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-99
SLIDE 99

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-100
SLIDE 100

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-101
SLIDE 101

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-102
SLIDE 102

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1) = f(m,n)+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-103
SLIDE 103

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1) = f(m,n)+1 = f(a,b)+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-104
SLIDE 104

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1) = f(m,n)+1 = f(a,b)+1 = f(a−1,b+1),

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-105
SLIDE 105

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1) = f(m,n)+1 = f(a,b)+1 = f(a−1,b+1), and by induction hypothesis:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-106
SLIDE 106

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1) = f(m,n)+1 = f(a,b)+1 = f(a−1,b+1), and by induction hypothesis: m−1 = a−1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-107
SLIDE 107

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1) = f(m,n)+1 = f(a,b)+1 = f(a−1,b+1), and by induction hypothesis: m−1 = a−1 and n+1 = b+1.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-108
SLIDE 108

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1) = f(m,n)+1 = f(a,b)+1 = f(a−1,b+1), and by induction hypothesis: m−1 = a−1 and n+1 = b+1. Hence (m,n) = (a,b)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-109
SLIDE 109

logo1 Finite vs. Infinite Countability Examples

Proof (injectivity, concl.). Induction step (m−1) → m: Let f(m,n) = f(a,b). WLOG m,a = 1. Then f(m−1,n+1) = f(m,n)+1 = f(a,b)+1 = f(a−1,b+1), and by induction hypothesis: m−1 = a−1 and n+1 = b+1. Hence (m,n) = (a,b) and f is injective.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-110
SLIDE 110

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-111
SLIDE 111

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-112
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logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1:

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 113

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 114

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 115

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 116

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 117

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-118
SLIDE 118

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-119
SLIDE 119

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 120

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 121

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 122

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 123

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 124

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

= 1 2(n+1)n+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-125
SLIDE 125

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

= 1 2(n+1)n+1 = 1 2(n−1)n+n+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 126

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

= 1 2(n+1)n+1 = 1 2(n−1)n+n+1 = 1 2

  • (1)+(n)−1
  • (1)+(n)−2
  • +(n)+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 127

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

= 1 2(n+1)n+1 = 1 2(n−1)n+n+1 = 1 2

  • (1)+(n)−1
  • (1)+(n)−2
  • +(n)+1

= f(1,n)+1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 128

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

= 1 2(n+1)n+1 = 1 2(n−1)n+n+1 = 1 2

  • (1)+(n)−1
  • (1)+(n)−2
  • +(n)+1

= f(1,n)+1 = k +1.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 129

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

= 1 2(n+1)n+1 = 1 2(n−1)n+n+1 = 1 2

  • (1)+(n)−1
  • (1)+(n)−2
  • +(n)+1

= f(1,n)+1 = k +1. Either way, k +1 has a preimage under f

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 130

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

= 1 2(n+1)n+1 = 1 2(n−1)n+n+1 = 1 2

  • (1)+(n)−1
  • (1)+(n)−2
  • +(n)+1

= f(1,n)+1 = k +1. Either way, k +1 has a preimage under f, and f is surjective.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 131

logo1 Finite vs. Infinite Countability Examples

Proof (surjectivity). We prove by induction on k that for every k ∈ N there are (m,n) ∈ N×N so that f(m,n) = k. Base step k = 1: f(1,1) = 1 2(1+1−1)(1+1−2)+1 = 1. Induction step k → k +1: Let f(m,n) = k. If m > 1, then f(m−1,n+1) = f(m,n)+1 = k +1. If m = 1, then f(n+1,1) = 1 2

  • (n+1)+(1)−1
  • (n+1)+(1)−2
  • +1

= 1 2(n+1)n+1 = 1 2(n−1)n+n+1 = 1 2

  • (1)+(n)−1
  • (1)+(n)−2
  • +(n)+1

= f(1,n)+1 = k +1. Either way, k +1 has a preimage under f, and f is surjective.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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logo1 Finite vs. Infinite Countability Examples

Definition.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 133

logo1 Finite vs. Infinite Countability Examples

  • Definition. Two sets A and B are called disjoint iff A∩B = /

0.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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logo1 Finite vs. Infinite Countability Examples

  • Definition. Two sets A and B are called disjoint iff A∩B = /
  • 0. A

family {Ci}i∈I is called pairwise disjoint iff for all i = j we have Ci ∩Cj = / 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 135

logo1 Finite vs. Infinite Countability Examples

  • Definition. Two sets A and B are called disjoint iff A∩B = /
  • 0. A

family {Ci}i∈I is called pairwise disjoint iff for all i = j we have Ci ∩Cj = / 0. Theorem.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 136

logo1 Finite vs. Infinite Countability Examples

  • Definition. Two sets A and B are called disjoint iff A∩B = /
  • 0. A

family {Ci}i∈I is called pairwise disjoint iff for all i = j we have Ci ∩Cj = / 0.

  • Theorem. Countable unions of countable sets are countable.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 137

logo1 Finite vs. Infinite Countability Examples

Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 138

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 139

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1

Cj.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-140
SLIDE 140

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 141

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 142

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 143

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N. “⊇”:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-144
SLIDE 144

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N. “⊇”: Let x ∈

α

  • n=1

Cn

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-145
SLIDE 145

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N. “⊇”: Let x ∈

α

  • n=1

Cn and let n ∈ N be the smallest natural number so that x ∈ Cn.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-146
SLIDE 146

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N. “⊇”: Let x ∈

α

  • n=1

Cn and let n ∈ N be the smallest natural number so that x ∈ Cn. Then x ∈ Bn

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-147
SLIDE 147

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N. “⊇”: Let x ∈

α

  • n=1

Cn and let n ∈ N be the smallest natural number so that x ∈ Cn. Then x ∈ Bn, which proves “⊇.”

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-148
SLIDE 148

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N. “⊇”: Let x ∈

α

  • n=1

Cn and let n ∈ N be the smallest natural number so that x ∈ Cn. Then x ∈ Bn, which proves “⊇.” The Bn are pairwise disjoint

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-149
SLIDE 149

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N. “⊇”: Let x ∈

α

  • n=1

Cn and let n ∈ N be the smallest natural number so that x ∈ Cn. Then x ∈ Bn, which proves “⊇.” The Bn are pairwise disjoint: If m < n, then Bm = Cm \

m−1

  • j=1

Cj ⊆ Cm

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-150
SLIDE 150

logo1 Finite vs. Infinite Countability Examples

  • Proof. Let {Cn}α

n=1 with α ∈ N∪{∞} be a countable family of

countable sets. Let Bn := Cn \

n−1

  • j=1
  • Cj. We claim

α

  • n=1

Bn =

α

  • n=1

Cn. “⊆”: Follows from Bn ⊆ Cn for all n ∈ N. “⊇”: Let x ∈

α

  • n=1

Cn and let n ∈ N be the smallest natural number so that x ∈ Cn. Then x ∈ Bn, which proves “⊇.” The Bn are pairwise disjoint: If m < n, then Bm = Cm \

m−1

  • j=1

Cj ⊆ Cm and Bn = Cn \

n−1

  • j=1

Cj ⊆ Cn \Cm.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 151

logo1 Finite vs. Infinite Countability Examples

Proof (concl.).

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 152

logo1 Finite vs. Infinite Countability Examples

Proof (concl.). For each n let Bn =

  • bk

n : k ∈ In

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 153

logo1 Finite vs. Infinite Countability Examples

Proof (concl.). For each n let Bn =

  • bk

n : k ∈ In

  • , where In = N
  • r In = {1,...,mn}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

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SLIDE 154

logo1 Finite vs. Infinite Countability Examples

Proof (concl.). For each n let Bn =

  • bk

n : k ∈ In

  • , where In = N
  • r In = {1,...,mn}. Then f(n,k) := bk

n is a bijective function

between

  • (n,k) ∈ N×N : n ≤ α,k ∈ In
  • ⊆ N×N and

α

  • n=1

Bn =

α

  • n=1

Cn.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-155
SLIDE 155

logo1 Finite vs. Infinite Countability Examples

Proof (concl.). For each n let Bn =

  • bk

n : k ∈ In

  • , where In = N
  • r In = {1,...,mn}. Then f(n,k) := bk

n is a bijective function

between

  • (n,k) ∈ N×N : n ≤ α,k ∈ In
  • ⊆ N×N and

α

  • n=1

Bn =

α

  • n=1
  • Cn. Hence

α

  • n=1

Cn is countable.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets

slide-156
SLIDE 156

logo1 Finite vs. Infinite Countability Examples

Proof (concl.). For each n let Bn =

  • bk

n : k ∈ In

  • , where In = N
  • r In = {1,...,mn}. Then f(n,k) := bk

n is a bijective function

between

  • (n,k) ∈ N×N : n ≤ α,k ∈ In
  • ⊆ N×N and

α

  • n=1

Bn =

α

  • n=1
  • Cn. Hence

α

  • n=1

Cn is countable.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Countable Sets