Cardinality of infinite sets - University of Kent...Cardinality of infinite sets Counting a...
Transcript of Cardinality of infinite sets - University of Kent...Cardinality of infinite sets Counting a...
Cardinality of infinite sets
Counting afinite setA is done by aone to one relationbetween the elements ofA and those of asuitable subset ofN:
. – p.1/30
Cardinality of infinite sets
Counting afinite setA is done by aone to one relationbetween the elements ofA and those of asuitable subset ofN:
α
β
γ
x
y
z
b
u
{ 1, 2, 3, 4, 5, 6, 7, 8}
⇒ |A| = 8.
. – p.1/30
Cardinality of infinite sets
Counting afinite setA is done by aone to one relationbetween the elements ofA and those of asuitable subset ofN:
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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Cardinality of infinite sets
Two setsA andB have the samecardinality, if there is afunction
f : A → B, a 7→ b, which is
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Cardinality of infinite sets
Two setsA andB have the samecardinality, if there is afunction
f : A → B, a 7→ b, which is
one to one( or "injective") and
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Cardinality of infinite sets
Two setsA andB have the samecardinality, if there is afunction
f : A → B, a 7→ b, which is
one to one( or "injective") and‘onto’ (or "surjective")
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Cardinality of infinite sets
Two setsA andB have the samecardinality, if there is afunction
f : A → B, a 7→ b, which is
one to one( or "injective") and‘onto’ (or "surjective")such a function is called "bijective".
. – p.2/30
Cardinality of infinite sets
Two setsA andB have the samecardinality, if there is afunction
f : A → B, a 7→ b, which is
one to one( or "injective") and‘onto’ (or "surjective")such a function is called "bijective".
"injective" means:a 6= a′ ⇒ f(a) 6= f(a′)."different inputs have different outputs."
. – p.2/30
Cardinality of infinite sets
Two setsA andB have the samecardinality, if there is afunction
f : A → B, a 7→ b, which is
one to one( or "injective") and‘onto’ (or "surjective")such a function is called "bijective".
"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
Cardinality of infinite sets
Example
This function isnot injective
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x
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b
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{ 1, 2, 3, 4, 5 }
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Cardinality of infinite sets
Example
This function isnot surjective
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b
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{ }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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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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. – p.11/30
Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
1. f : R → R+ : x 7→ x2
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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
1. f : R → R+ : x 7→ x2
2. f : Z → Z : x 7→ 2x
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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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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Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
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
. – p.12/30
Injective, surjective, bijective
Notice that a functionf : X → Y is defined including itsdomainX and its codomainY .
. – p.13/30
Injective, surjective, bijective
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?
. – p.13/30
Cardinality of infinite sets
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|.
. – p.14/30
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...
. – p.15/30
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.
. – p.15/30
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
Proof: Can assumeB = N andA infinite.
Seta1 := min(A); a2 := min(A\{a1}),a3 := min(A\{a1, a2}) · · ·
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Theorem
A ⊆ B andB countable⇒ A countable
Proof: Can assumeB = N andA infinite.
Seta1 := min(A); a2 := min(A\{a1}),a3 := min(A\{a1, a2}) · · ·
ak := min(A\{a1, a2, · · · , ak−1}).
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Theorem
A ⊆ B andB countable⇒ A countable
Proof: Can assumeB = N andA infinite.
Seta1 := min(A); a2 := min(A\{a1}),a3 := min(A\{a1, a2}) · · ·
ak := min(A\{a1, a2, · · · , ak−1}).
a1 < a2 < a3 · · ·
The mapf : N → A : k 7→ ak is bijective.(check this!)
. – p.16/30
True or false, explain
1. A countable andB countable⇒ A × B is countable.
2. Z × Z is countable
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True or false, explain
1. A countable andB countable⇒ A × B is countable.
2. Z × Z is countable
3. Z × Z × Z is countable
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True or false, explain
1. A countable andB countable⇒ A × B is countable.
2. Z × Z is countable
3. Z × Z × Z is countable
4. Q is countable.
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True or false, explain
1. A countable andB countable⇒ A × B is countable.
2. Z × Z is countable
3. Z × Z × Z is countable
4. Q is countable.
5. |(0, 1)| = |R|.
. – p.17/30