32 KiB
title | TARGET DECK | FILE TAGS | tags | |
---|---|---|---|---|
Cardinality | Obsidian::STEM | set::cardinality |
|
Equinumerosity
We say set A
is equinumerous to set B
, written (A \approx B
) if and only if there exists a set/functions#Injections function from A
set/functions#Surjections B
.
%%ANKI
Basic
Suppose A
is equinumerous to B
. How does Enderton denote this?
Back: A \approx B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does it mean for A
to be equinumerous to B
?
Back: There exists a bijection between A
and B
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A \approx B
. Then what must exist?
Back: A bijection between A
and B
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose there exists a one-to-one function F
from A
into B
. When does this imply A \approx B
?
Back: When F
is also onto B
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose there exists a function F
from A
onto B
. When does this imply A \approx B
?
Back: When F
is also one-to-one.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose there exists a one-to-one function F
from A
onto B
. When does this imply A \approx B
?
Back: Always, by definition.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Power Sets
No set is equinumerous to its set/index#Power Set Axiom. This is typically shown using a diagonalization argument.
%%ANKI Basic What basic set operation produces a new set the original isn't equinumerous to? Back: The power set operation. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What kind of argument is typically made to prove no set is equinumerous to its power set? Back: A diagonalization argument. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic Who is attributed the discovery of the diagonalization argument? Back: Georg Cantor. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let g \colon A \rightarrow \mathscr{P}A
. Using a diagonalization argument, what set is not in \mathop{\text{ran}}(g)
?
Back: \{ x \in A \mid x \not\in g(x) \}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let g \colon A \rightarrow \mathscr{P}A
. Why isn't B = \{x \in A \mid x \not\in g(x) \}
in \mathop{\text{ran}}(g)
?
Back: For all x \in A
, x \in B \Leftrightarrow x \not\in g(x)
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Equivalence Concept
For any sets A
, B
, and C
:
A \approx A
;- if
A \approx B
, thenB \approx A
; - if
A \approx B
andB \approx C
, thenA \approx C
.
Notice though that \{ \langle A, B \rangle \mid A \approx B \}
is not an equivalence relation since the equivalence concept of equinumerosity concerns all sets.
%%ANKI
Basic
Concisely state the equivalence concept of equinumerosity in Zermelo-Fraenkel set theory.
Back: For all sets A
, B
, and C
:
A \approx A
;A \approx B \Rightarrow B \approx A
;A \approx B \land B \approx C \Rightarrow A \approx C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Concisely state the equivalence concept of equinumerosity in von Neumann-Bernays set theory.
Back: Class \{ \langle A, B \rangle \mid A \approx B \}
is reflexive on the class of all sets, symmetric, and transitive.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the reflexive property of equinumerosity in FOL?
Back: \forall A, A \approx A
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the symmetric property of equinumerosity in FOL?
Back: \forall A, B, A \approx B \Rightarrow B \approx A
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the transitive property of equinumerosity in FOL?
Back: \forall A, B, C, A \approx B \land B \approx C \Rightarrow A \approx C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Is \{ \langle A, B \rangle \mid A \approx B \}
a set?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why isn't \{ \langle A, B \rangle \mid A \approx B \}
a set?
Back: Because then the field of this "relation" would be a set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Is \{ \langle A, B \rangle \mid A \approx B \}
an equivalence relation?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why isn't \{ \langle A, B \rangle \mid A \approx B \}
an equivalence relation?
Back: Because then the field of this "relation" would be a set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Finiteness
A set is finite if and only if it is equinumerous to a natural-numbers. Otherwise it is infinite.
%%ANKI Basic How does Enderton define a finite set? Back: As a set equinumerous to some natural number. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic How does Enderton define an infinite set? Back: As a set not equinumerous to any natural number. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Is n \in \omega
a finite set?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why isn't n \in \omega
a finite set?
Back: N/A. It is.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Is \omega
a finite set?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why isn't \omega
a finite set?
Back: There is no natural number equinumerous to \omega
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Pigeonhole Principle
No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself.
Likewise, any set equinumerous to a proper subset of itself must be infinite.
%%ANKI
Basic
How does Enderton state the pigeonhole principle for \omega
?
Back: No natural number is equinumerous to a proper subset of itself.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic How does Enderton state the pigeonhole principle for finite sets? Back: No finite set is equinumerous to a proper subset of itself. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m \in n \in \omega
. What principle precludes m \approx n
?
Back: The pigeonhole principle.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S
be a set and n \in \omega
such that S \approx n
. For m \in \omega
, when might S \approx m
?
Back: Only if m = n
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the generalization of the pigeonhole principle for \omega
?
Back: The pigeonhole principle for finite sets.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the specialization of the pigeonhole principle for finite sets?
Back: The pigeonhole principle for \omega
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What name is given to the following theorem? \text{No finite set is equinumerous to a proper subset of itself.}
Back: The pigeonhole principle.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S
be a finite set and f \colon S \rightarrow S
be injective. Is f
a bijection?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S
be a finite set and f \colon S \rightarrow S
be injective. Why must f
be surjective?
Back: Otherwise f
is a bijection between S
and a proper subset of S
, a contradiction.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S
be a finite set and f \colon S \rightarrow S
be surjective. Is f
a bijection?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S
be a finite set and f \colon S \rightarrow S
be surjective. Why must f
be injective?
Back: Otherwise f
is a bijection between a proper subset of S
and S
, a contradiction.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What does the contrapositive of the pigeonhole principle state? Back: Any set equinumerous to a proper subset of itself is infinite. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What general strategy is used to prove \omega
is an infinite set?
Back: Prove \omega
is equinumerous to a proper subset of itself.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Cardinal Numbers
A cardinal number is a set that is \mathop{\text{card}} A
for some set A
. The set \mathop{\text{card}} A
is defined such that
- For any sets
A
andB
,\mathop{\text{card}}A = \mathop{\text{card}}B
iffA \approx B
. - For a finite set
A
,\mathop{\text{card}}A
is the natural numbern
for whichA \approx n
.
%%ANKI
Basic
How is the cardinal number of set A
denoted?
Back: As \mathop{\text{card}} A
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A
is finite. What does \mathop{\text{card}} A
evaluate to?
Back: The unique n \in \omega
such that A \approx n
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider n \in \omega
. What does \mathop{\text{card}} n
evaluate to?
Back: n
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose a
, b
, and c
are distinct objects. What does \mathop{\text{card}} \{a, b, c\}
evaluate to?
Back: 3
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What does Enderton refer to by the "process called 'counting'"? Back: Choosing a one-to-one correspondence between two sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
A {cardinal number} is a set that is {\mathop{\text{card} } A
} for some set A
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How do cardinal numbers relate to equinumerosity?
Back: For any sets A
and B
, \mathop{\text{card}} A = \mathop{\text{card}} B
iff A \approx B
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
According to Enderton, what is the "essential demand" for defining cardinal numbers?
Back: Defining cardinal numbers such that for any sets A
and B
, \mathop{\text{card}} A = \mathop{\text{card}} B
iff A \approx B
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What name is given to \mathop{\text{card}} \omega
?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Who is attributed the assignment \mathop{\text{card}} \omega = \aleph_0
?
Back: Georg Cantor.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
If one set A
of cardinality \kappa
is finite, then all of them are. In this case \kappa
is a finite cardinal. Otherwise \kappa
is an infinite cardinal.
%%ANKI
Basic
How many sets A
exist such that \mathop{\text{card}} A = 0
?
Back: 1
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How many sets A
exist such that \mathop{\text{card}} A = n^+
for some n \in \omega
?
Back: An infinite many.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \omega
. When is \{X \mid \mathop{\text{card}} X = n\}
a set?
Back: When n = 0
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \omega
. When is \{X \mid \mathop{\text{card}} X = n\}
a class?
Back: Always.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What class can we construct to prove \{X \mid \mathop{\text{card}} X = 1\}
is not a set?
Back: \bigcup\, \{\{X\} \mid X \text{ is a set} \}
, i.e. the union of all singleton sets.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is a finite cardinal?
Back: A cardinal number equal to \mathop{\text{card}} A
for some finite set A
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is an infinite cardinal?
Back: A cardinal number equal to \mathop{\text{card}} A
for some infinite set A
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
The finite cardinals are exactly what more basic set?
Back: \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What set does \aleph_0
refer to?
Back: \mathop{\text{card}} \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the "smallest" infinite cardinal?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let C \subseteq A
where A \approx n
for some n \in \omega
. What does \mathop{\text{card}} C
evaluate to?
Back: A natural number m
such that m \underline{\in} n
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let C \subset A
where A \approx n
for some n \in \omega
. What does \mathop{\text{card}} C
evaluate to?
Back: A natural number m
such that m \in n
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is proposition "any subset of a finite set is finite" expressed in FOL?
Back: \forall n \in \omega, \forall A \approx n, \forall B \subseteq A, \exists m \in n, B \approx m
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is the following more succinctly stated? \forall n \in \omega, \forall A \approx n, \forall B \subseteq A, \exists m \in n, B \approx m$$
Back: Any subset of a finite set is finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose sets A
and B
are finite. When is A \cup B
infinite?
Back: N/A. The union of two finite sets is always finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
s.t. A \approx m
and B \approx n
. What is the largest value \mathop{\text{card}}(A \cup B)
can evaluate to?
Back: m + n
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
s.t. A \approx m
and B \approx n
. What is the smallest value \mathop{\text{card}}(A \cup B)
can evaluate to?
Back: \mathop{\text{max}}(m, n)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
s.t. A \approx m
and B \approx n
. When does \mathop{\text{card}}(A \cup B) = m + n
?
Back: When A
and B
are disjoint.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
s.t. A \approx m
and B \approx n
. When does \mathop{\text{card}}(A \cup B) = m
?
Back: When B \subseteq A
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose sets A
and B
are finite. When is A \cap B
infinite?
Back: N/A. The intersection of two finite sets is always finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
s.t. A \approx m
and B \approx n
. What is the largest value \mathop{\text{card}}(A \cap B)
can evaluate to?
Back: \mathop{\text{min}}(m, n)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
s.t. A \approx m
and B \approx n
. What is the smallest value \mathop{\text{card}}(A \cap B)
can evaluate to?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose sets A
and B
are finite. When is A \times B
infinite?
Back: N/A. The Cartesian product of two finite sets is always finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Addition
Let \kappa
and \lambda
be any cardinal numbers. Then \kappa + \lambda = \mathop{\text{card}}(K \cup L)
, where K
and L
are any disjoint sets of cardinality \kappa
and \lambda
, respectively.
%%ANKI
Basic
Let \kappa
and \lambda
be any cardinal numbers. How is \kappa + \lambda
defined?
Back: As \mathop{\text{card}}(K \cup L)
where K
and L
are disjoint sets with cardinality \kappa
and \lambda
, respectively.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K
and L
be disjoint sets. What does \mathop{\text{card}}(K \cup L)
evaluate to?
Back: As \kappa + \lambda
where \kappa = \mathop{\text{card}} K
and \lambda = \mathop{\text{card}} L
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \mathop{\text{card}}(K) = \kappa
and \mathop{\text{card}}(L) = \lambda
. What is necessary for \mathop{\text{card}}(K \cup L) = \kappa + \lambda
?
Back: That K
and L
are disjoint.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Cloze {Addition} of cardinal numbers is defined in terms of the {union} of sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How do we prove 2 + 2 = 4
using the recursion theorem?
Back: By proving A_2(2) = 2^{++} = 4
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How do we prove 2 + 2 = 4
using cardinal numbers?
Back: By proving for disjoint sets K \approx 2
and L \approx 2
, that K \cup L \approx 4
holds.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
. What does m + n
evaluate to in terms of cardinal numbers?
Back: \mathop{\text{card}}((m \times \{0\}) \cup (n \times \{1\}))
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What cardinal number does 0 + \aleph_0
evaluate to?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Expression 0 + \aleph_0
corresponds to the cardinality of what set?
Back: \varnothing \cup \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \omega
. What cardinal number does n^+ + \aleph_0
evaluate to?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \omega
. Expression n + \aleph_0
corresponds to the cardinality of what set?
Back: (n \times \{0\}) \cup (\omega \times \{1\})
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What cardinal number does \aleph_0 + \aleph_0
evaluate to?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Expression \aleph_0 + \aleph_0
corresponds to the cardinality of what set?
Back: (\omega \times \{0\}) \cup (\omega \times \{1\})
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa
be a cardinal number. What cardinal number does \kappa + 0
evaluate to?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Multiplication
Let \kappa
and \lambda
be any cardinal numbers. Then \kappa \cdot \lambda = \mathop{\text{card}}(K \times L)
, where K
and L
are any sets of cardinality \kappa
and \lambda
, respectively.
%%ANKI
Basic
Let \kappa
and \lambda
be any cardinal numbers. How is \kappa \cdot \lambda
defined?
Back: As \mathop{\text{card}}(K \times L)
where K
and L
are sets with cardinality \kappa
and \lambda
, respectively.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K
and L
be sets. What does \mathop{\text{card}}(K \times L)
evaluate to?
Back: As \kappa \cdot \lambda
where \kappa = \mathop{\text{card}} K
and \lambda = \mathop{\text{card}} L
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K \approx \kappa
and L \approx \lambda
. What is necessary for \mathop{\text{card}}(K \times L) \approx \kappa \cdot \lambda
?
Back: N/A. This is true by definition.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Cloze {Multiplication} of cardinal numbers is defined in terms of the {Cartesian product} of sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
such that A \approx m
and B \approx n
. What does \mathop{\text{card}}(A \times B)
evaluate to?
Back: m \cdot n
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How do we prove 2 \cdot 2 = 4
using the recursion theorem?
Back: By proving M_2(2) = 2 + 2 = 4
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How do we prove 2 \cdot 2 = 4
using cardinal numbers?
Back: By proving for sets K \approx 2
and L \approx 2
, that K \times L \approx 4
holds.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
. What does m \cdot n
evaluate to in terms of cardinal numbers?
Back: \mathop{\text{card}}(m \times n)
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What cardinal number does 0 \cdot \aleph_0
evaluate to?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Expression 0 \cdot \aleph_0
corresponds to the cardinality of what set?
Back: \varnothing \times \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \omega
. What cardinal number does n^+ \cdot \aleph_0
evaluate to?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \omega
. Expression n \cdot \aleph_0
corresponds to the cardinality of what set?
Back: n \times \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What cardinal number does \aleph_0 \cdot \aleph_0
evaluate to?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Expression \aleph_0 \cdot \aleph_0
corresponds to the cardinality of what set?
Back: \omega \times \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa
be a cardinal number. What cardinal number does \kappa \cdot 0
evaluate to?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa
be a cardinal number. What cardinal number does \kappa \cdot 1
evaluate to?
Back: \kappa
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Exponentiation
Let \kappa
and \lambda
be any cardinal numbers. Then \kappa^\lambda = \mathop{\text{card}}(^LK)
, where K
and L
are any sets of cardinality \kappa
and \lambda
, respectively.
%%ANKI
Basic
Let \kappa
and \lambda
be any cardinal numbers. How is \kappa^\lambda
defined?
Back: As \mathop{\text{card}}(^LK)
where K
and L
are sets with cardinality \kappa
and \lambda
, respectively.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K
and L
be sets. What does \mathop{\text{card}}(^LK)
evaluate to?
Back: As \kappa^\lambda
where \kappa = \mathop{\text{card}} K
and \lambda = \mathop{\text{card}} L
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K
and L
be sets. How is \mathop{\text{card}}(^KL)
expressed in terms of cardinal numbers?
Back: As \lambda^\kappa
where \kappa = \mathop{\text{card}} K
and \lambda = \mathop{\text{card}} L
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K \approx \kappa
and L \approx \lambda
. What is necessary for \mathop{\text{card}}(^LK) \approx \kappa^\lambda
?
Back: N/A. This is true by definition.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Cloze {Exponentiation} of cardinal numbers is defined in terms of the {set of functions} between sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How do we prove 2^2 = 4
using the recursion theorem?
Back: By proving E_2(2) = 2 \cdot 2 = 4
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How do we prove 2^2 = 4
using cardinal numbers?
Back: By proving for sets K \approx 2
and L \approx 2
, that ^LK \approx 4
holds.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega
. What does m^n
evaluate to in terms of cardinal numbers?
Back: \mathop{\text{card}}(^nm)
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What cardinal number does 0^{\aleph_0}
evaluate to?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Expression 0^{\aleph_0}
corresponds to the cardinality of what set?
Back: ^\omega \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa
be a nonzero cardinal number. What cardinal number does 0^\kappa
evaluate to?
Back: 0
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What cardinal number does 0^0
evaluate to?
Back: 1
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa
be a cardinal number. Expression 0^\kappa
corresponds to the cardinality of what set?
Back: ^K\varnothing
where \mathop{\text{card}} K = \kappa
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa
be a cardinal number. What cardinal number does \kappa^0
evaluate to?
Back: 1
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa
be a cardinal number. Expression \kappa^0
corresponds to the cardinality of what set?
Back: ^\varnothing K
where \mathop{\text{card}} K = \kappa
..
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Bibliography
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).