Cardinality of infinite sets - University of Kent...Cardinality of infinite sets Counting a finite set A is done by a one to one relation between the elements of Aand those of a

Cardinality of infinite sets

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Counting afinite setA is done by aone to one relationbetween the elements ofA and those of asuitable subset ofN:

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α

β

γ

x

y

z

b

u

{ 1, 2, 3, 4, 5, 6, 7, 8}

⇒ |A| = 8.

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α

β

γ

x

y

z

b

u

{ 1, 2, 3, 4, 5, 6, 7, 8}

⇒ |A| = 8.

In a similar way one can compare the sizes ofinfinite sets.

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Two setsA andB have the samecardinality, if there is afunction

f : A → B, a 7→ b, which is

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one to one( or "injective") and

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one to one( or "injective") and‘onto’ (or "surjective")

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one to one( or "injective") and‘onto’ (or "surjective")such a function is called "bijective".

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"injective" means:a 6= a′ ⇒ f(a) 6= f(a′)."different inputs have different outputs."

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"injective" means:a 6= a′ ⇒ f(a) 6= f(a′)."different inputs have different outputs."

"surjective" means:∀b ∈ B : ∃a ∈ A such thatb = f(a)."Every element ofB is ‘hit’ by the functionf ." . – p.2/30


Example

This function isnot injective

α

β

γ

x

y

z

b

u

{ 1, 2, 3, 4, 5 }

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Example

This function isnot surjective

α

β

γ

x

y

z

b

u

{ }1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

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Injective, surjective, bijective

Which of the following functions are injective, surjective,bijective or neither?

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1. f : R → R+ : x 7→ x2

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1. f : R → R+ : x 7→ x2

2. f : Z → Z : x 7→ 2x

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1. f : R → R+ : x 7→ x2

2. f : Z → Z : x 7→ 2x

3. f : (−1, 1) → R : x 7→ x

1−|x|

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1. f : R → R+ : x 7→ x2

2. f : Z → Z : x 7→ 2x

3. f : (−1, 1) → R : x 7→ x

1−|x|

4. f : R\{2} → R\{1} : x 7→ x+1

x−2

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Notice that a functionf : X → Y is defined including itsdomainX and its codomainY .

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Notice that a functionf : X → Y is defined including itsdomainX and its codomainY .

f1 : R → R : x 7→ x2

f2 : R+ → R+ : x 7→ x2

f3 : R → R+ : x 7→ x2

are three different functions. Which of them is injective,surjective, bijective or neither?

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Example

The function

f : Z → 2Z := {z ∈ Z | z is even}, m 7→ 2m

is injectiveandsujective.

⇒: |Z| = |2Z|.

A setA is calledcountableif A is finite or if |A| = |N|.

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Countable sets

Z is countable

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Countable sets

Z is countable

f : N → Z, 2k − 1 7→ −k, 2k 7→ k − 1, k = 1, 2, · · · .

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Countable sets

Z is countable

f : N → Z, 2k − 1 7→ −k, 2k 7→ k − 1, k = 1, 2, · · · .

N × N is countable

12345678910...

1 2 3 4 5 6 7 8 9 10 11...

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Countable sets

Z is countable

f : N → Z, 2k − 1 7→ −k, 2k 7→ k − 1, k = 1, 2, · · · .

N × N is countable

12345678910...

1 2 3 4 5 6 7 8 9 10 11...

A ⊆ B andB countable⇒ A countable.

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Countable sets

Z is countable

f : N → Z, 2k − 1 7→ −k, 2k 7→ k − 1, k = 1, 2, · · · .

N × N is countable

12345678910...

1 2 3 4 5 6 7 8 9 10 11...

A ⊆ B andB countable⇒ A countable.

If I is a countable set and for everyi ∈ I Ai is a countable setthen∪i∈I Ai is countable.

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Theorem

A ⊆ B andB countable⇒ A countable

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Theorem


Proof: Can assumeB = N andA infinite.

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Theorem



Seta1 := min(A); a2 := min(A\{a1}),a3 := min(A\{a1, a2}) · · ·

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Theorem




ak := min(A\{a1, a2, · · · , ak−1}).

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Theorem




ak := min(A\{a1, a2, · · · , ak−1}).

a1 < a2 < a3 · · ·

The mapf : N → A : k 7→ ak is bijective.(check this!)

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True or false, explain

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1. A countable andB countable⇒ A × B is countable.

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2. Z × Z is countable

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3. Z × Z × Z is countable

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4. Q is countable.

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4. Q is countable.

5. |(0, 1)| = |R|.

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4. Q is countable.

5. |(0, 1)| = |R|.

all true.

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Cardinality of infinite sets - University of Kent...Cardinality of infinite sets Counting a finite set A is done by a one to one relation between the elements of Aand those of a