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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.
. – p.1/30
Cardinality of infinite sets
Two setsA andB have the samecardinality, if there is afunction
f : A → B, a 7→ b, which is
. – 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
. – 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")
. – 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".
. – 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
α
β
γ
x
y
z
b
u
{ 1, 2, 3, 4, 5 }
. – p.3/30
Cardinality of infinite sets
Example
This function isnot surjective
α
β
γ
x
y
z
b
u
{ }1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
. – p.4/30
Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
. – p.5/30
Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
.
.
.
.
.
.
.
.
.
.
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.
.
. – p.6/30
Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
.
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.
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.
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.. – p.7/30
Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
.
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. – p.8/30
Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
.
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. – p.9/30
Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
.
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.
.
.
.
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. – p.10/30
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?
. – p.12/30
Injective, surjective, bijective
Which of the following functions are injective, surjective,bijective or neither?
1. f : R → R+ : x 7→ x2
. – p.12/30
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
. – p.12/30
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|
. – p.12/30
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.
. – p.15/30
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}) · · ·
. – p.16/30
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}).
. – p.16/30
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
. – p.17/30
True or false, explain
1. A countable andB countable⇒ A × B is countable.
2. Z × Z is countable
3. Z × Z × Z is countable
. – p.17/30
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.
. – p.17/30
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