58 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.
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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%%
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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%%
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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%%
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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%%
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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.
For any set A, ^A2 \approx \mathscr{P}(A).
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Basic
What kind of argument is typically used to show A \not\approx {\mathscr{P}(A)}?
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%%
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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%%
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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%%
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Basic
For any set A, \mathscr{P}(A) is equinumerous to what set of functions?
Back: ^A2
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Let A be any set. How is bijection H \colon \,^A2 \rightarrow \mathscr{P}(A) typically defined?
Back: H(f) = \{a \in A \mid f(a) = 1\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
For any set A, ^A2 is equinumerous to what of A?
Back: Its powerset.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What kind of argument is typically used to show A \not\approx {^A2}?
Back: A diagonalization argument.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
For any set A, what is the cardinality of its powerset?
Back: 2^{\mathop{\text{card}}A}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What does \mathop{\text{card}} \mathscr{P}(A) evaluate to?
Back: 2^{\mathop{\text{card}}A}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
The cardinality of what set equals 2^{\mathop{\text{card}}A}?
Back: \mathscr{P}(A)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What does \mathop{\text{card}} \mathscr{P}(\omega) evaluate to?
Back: 2^{\aleph_0}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Why is the "power set" named the way it is?
Back: Because \mathop{\text{card}} \mathscr{P}(A) equals 2 to the power of \mathop{\text{card}} A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
How do we know \aleph_0 \not\approx 2^{\aleph_0} holds?
Back: No set is equinumerous to its power set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
For any cardinal number \kappa, how do we know \kappa \not\approx 2^\kappa?
Back: No set is equinumerous to its power set.
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 BandB \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.
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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 CReference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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Basic
Is n \in \omega a finite set?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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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%%
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Basic
Is \omega a finite set?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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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.
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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
AandB,\mathop{\text{card}}A = \mathop{\text{card}}BiffA \approx B. - For a finite set
A,\mathop{\text{card}}Ais the natural numbernfor whichA \approx n.
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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%%
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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%%
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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%%
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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%%
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Cloze
A {cardinal number} is denoted as {\mathop{\text{card} } A} for some set A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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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%%
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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%%
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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.
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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Basic
What is the "smallest" infinite cardinal?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
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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%%
%%ANKI
Basic
Suppose sets A and B are finite. When is ^BA infinite?
Back: N/A. The set of functions from one finite set to another 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: \mathop{\text{card}}K + \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%%
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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: \kappa
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Let \kappa and \lambda be cardinal numbers. Does \kappa + \lambda = \lambda + \kappa?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. Why does \kappa + \lambda = \lambda + \kappa?
Back: Because the union of sets is commutative.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa, \lambda, and \mu be cardinal numbers. Does \kappa + (\lambda + \mu) = (\kappa + \lambda) + \mu?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa, \lambda, and \mu be cardinal numbers. Why does \kappa + (\lambda + \mu) = (\kappa + \lambda) + \mu?
Back: Because the union of sets is associative.
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: \mathop{\text{card}}K \cdot \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 \times L) = \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%%
%%ANKI
Cloze
For any cardinal number \kappa, addition's {\kappa + \kappa} equals multiplication's {2 \cdot \kappa}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. Does \kappa \cdot \lambda = \lambda \cdot \kappa?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. Why does \kappa \cdot \lambda = \lambda \cdot \kappa?
Back: Because K \times L \approx L \times K for any sets K and L.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa, \lambda, and \mu be cardinal numbers. Does \kappa \cdot (\lambda \cdot \mu) = (\kappa \cdot \lambda) \cdot \mu?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa, \lambda, and \mu be cardinal numbers. Why does \kappa \cdot (\lambda \cdot \mu) = (\kappa \cdot \lambda) \cdot \mu?
Back: Because K \times (L \times M) \approx (K \times L) \times M for any sets K, L, and M.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa, \lambda, and \mu be cardinal numbers. What does the distributive property state?
Back: \kappa \cdot (\lambda + \mu) = (\kappa \cdot \lambda) + (\kappa \cdot \mu)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa, \lambda, and \mu be cardinal numbers. Why does \kappa \cdot (\lambda + \mu) = (\kappa \cdot \lambda) + (\kappa \cdot \mu)?
Back: Because the Cartesian product distributes over the union operation.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. What does \kappa \cdot (\lambda + 1) evaluate to?
Back: \kappa \cdot \lambda + \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: To \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 \mathop{\text{card}}K = \kappa and \mathop{\text{card}} L = \lambda. What is necessary for \mathop{\text{card}}(^LK) = \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%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. Does \kappa ^ \lambda = \lambda ^ \kappa?
Back: Not necessarily.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Let \kappa, \lambda, and \mu be cardinal numbers. Then \kappa^{\lambda + \mu} = {\kappa^\lambda \cdot \kappa^\mu}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Let \kappa, \lambda, and \mu be cardinal numbers. Then (\kappa \cdot \lambda)^\mu = {\kappa^\mu \cdot \lambda^\mu}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Let \kappa, \lambda, and \mu be cardinal numbers. Then (\kappa^\lambda)^\mu = {\kappa^{\lambda \cdot \mu}}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa, \lambda, and \mu be cardinal numbers. What \lambda-calculus concept does (\kappa^\lambda)^\mu = \kappa^{\lambda \cdot \mu} embody?
Back: Currying.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. Rewrite \kappa^{\lambda + 1} without using addition.
Back: \kappa^\lambda \cdot \kappa
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa be a cardinal number. How is the factorial of \kappa denoted?
Back: \kappa !
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa be a cardinal number. How is the factorial of \kappa defined?
Back: As \mathop{\text{card}} \{ f \mid f \text{ is a permutation of } K\} for some \mathop{\text{card}} K = \kappa.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Ordering
A set A is dominated by a set B, written A \preceq B, if and only if there is a one-to-one function from A into B. In other words, A \preceq B if and only if A is equinumerous to some subset of B. Then \mathop{\text{card}}A \leq \mathop{\text{card}}B \text{ if and only if } A \preceq B.
Furthermore, \mathop{\text{card}}A < \mathop{\text{card}}B \text{ if and only if } A \preceq B \text{ and } A \not\approx B.
%%ANKI
Basic
How do we denote that set A is dominated by set B?
Back: A \preceq B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How do we denote that set A is strictly dominated by set B?
Back: A \prec B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A \preceq B. Then what must exist by definition?
Back: A one-to-one function from A into B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A \prec B. Then what must exist by definition?
Back: A one-to-one function from A into B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A \preceq B. Then what must A be equinumerous to?
Back: A subset of B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A \preceq B. Then what must A not be equinumerous to?
Back: N/A. There is no restriction here.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A \prec B. Then what must A be equinumerous to?
Back: A subset of B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A \prec B. Then what must A not be equinumerous to?
Back: B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does A \preceq B denote?
Back: That A is dominated by B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does A \prec B denote?
Back: That A is strictly dominated by B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
In terms of sets, how do we expand expression A \preceq B using FOL?
Back: \exists C, C \subseteq B \land A \approx C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
In terms of sets, how do we expand expression A \prec B using FOL?
Back: A \not\approx B \land \exists C, C \subseteq B \land A \approx C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. How is \kappa \leq \lambda defined?
Back: As K \preceq L for sets satisfying \mathop{\text{card}}K = \kappa and \mathop{\text{card}} L = \lambda.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. How is \kappa < \lambda defined?
Back: As K \preceq L and K \not\approx L for sets satisfying \mathop{\text{card}}K = \kappa and \mathop{\text{card}} L = \lambda.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
{\leq} on cardinal numbers corresponds to {\preceq} on sets.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
{<} on cardinal numbers corresponds to {\prec} on sets.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is \mathop{\text{card} }K \leq \mathop{\text{card} }L defined in terms of equinumerosity?
Back: \mathop{\text{card} }K \leq \mathop{\text{card} }L iff K is equinumerous to a subset of L.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is \mathop{\text{card} }K < \mathop{\text{card} }L defined in terms of equinumerosity?
Back: \mathop{\text{card} }K < \mathop{\text{card} }L iff K is equinumerous to a subset of L and K \not\approx L.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinal numbers. Restate the following in terms of sets: \kappa < \lambda \text{ iff } \kappa \leq \lambda \text{ and } \kappa \neq \lambda$$
Back: Given \mathop{\text{card}}K = \kappa and \mathop{\text{card}}L = \lambda, \mathop{\text{card}}K < \mathop{\text{card}}L iff K \preceq L and K \not\approx L.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K and L be sets. Restate the following in terms of cardinal numbers: \mathop{\text{card}}K < \mathop{\text{card}}L \text{ iff } K \preceq L \text{ and } K \not\approx L.
Back: Given \mathop{\text{card}}K = \kappa and \mathop{\text{card}}L = \lambda, \kappa < \lambda iff \kappa \leq \lambda and \kappa \neq \lambda.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K and L be sets. Why can't we use the following definition? \mathop{\text{card}} K \leq \mathop{\text{card}} L \text{ iff } \exists A \subseteq L, K \approx A$$
Back: N/A. This is a suitable definition.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K and L be sets. Why can't we use the following definition? \mathop{\text{card}} K < \mathop{\text{card}} L \text{ iff } \exists A \subset L, K \approx A$$
Back: Infinite sets may be equinumerous to a proper subset of themselves.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For any n \in \omega, why is n < \aleph_0?
Back: n \not\approx \omega and there exists an injective function f \colon n \rightarrow \omega.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For any cardinal number \kappa, why is \kappa < 2^\kappa?
Back: Assuming \mathop{\text{card}}K = \kappa, K \not\approx \mathscr{P}(K) and there exists an injective function f \colon K \rightarrow \mathscr{P}(K).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the smallest cardinal number?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What is the largest cardinal number? Back: N/A. There is no largest cardinal number. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa be a cardinal number. Does \kappa \leq \kappa hold true?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa be a cardinal number. Restate \kappa \leq \kappa in terms of sets.
Back: Let K be a set s.t. \mathop{\text{card}}K = \kappa. Then K \preceq K.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K be a set. Restate K \preceq K in terms of cardinal numbers.
Back: Assuming \mathop{\text{card}}K = \kappa, \kappa \leq \kappa.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why does \kappa \leq \kappa for any cardinal number \kappa?
Back: For set K s.t. \mathop{\text{card}}K = \kappa, K \preceq K.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For set K, which function most naturally proves K \preceq K?
Back: The identity function.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does it mean for cardinal numbers to obey transitivity?
Back: Let \kappa, \lambda, and \mu be cardinal numbers. If \kappa \leq \lambda and \lambda \leq \mu, then \kappa \leq \mu.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa, \lambda, and \mu be cardinal numbers. Restate the following in terms of sets: \text{if } \kappa \leq \lambda \text{ and } \lambda \leq \mu, \text{ then } \kappa \leq \mu$$
Back: Let K, L, and M be sets s.t. \mathop{\text{card}}K = \kappa, \mathop{\text{card}}L = \lambda, and \mathop{\text{card}}M = \mu. Then \text{if } K \preceq L \text{ and } L \preceq M, \text{ then } K \preceq M$$
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K, L, and M be sets. Restate the following in terms of cardinal numbers: \text{if } K \preceq L \text{ and } L \preceq M, \text{ then } K \preceq M$$
Back: Let \mathop{\text{card}}K = \kappa, \mathop{\text{card}}L = \lambda, and \mathop{\text{card}}M = \mu. Then \text{if } \kappa \leq \lambda \text{ and } \lambda \leq \mu, \text{ then } \kappa \leq \mu$$
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose K \preceq L and L \preceq M. Why must K \preceq M?
Back: There exist injective functions f \colon K \rightarrow L and g \colon L \rightarrow M. Then f \circ g is one-to-one from K to M.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Schröder-Bernstein Theorem
For any sets A and B, if A \preceq B and B \preceq A, then A \approx B.
%%ANKI
Basic
In terms of sets, what does the Schröder-Bernstein theorem state?
Back: For any sets A and B, if A \preceq B and B \preceq A, then A \approx B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
In terms of cardinal numbers, what does the Schröder-Bernstein theorem state?
Back: For any cardinal numbers \kappa and \lambda, if \kappa \leq \lambda and \lambda \leq \kappa, then \kappa = \lambda.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \kappa and \lambda be cardinals numbers. What name is given to the following conditional? \kappa \leq \lambda \land \lambda \leq \kappa \Rightarrow \kappa = \lambda$$
Back: The Schröder-Bernstein theorem.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let A and B be sets. What name is given to the following conditional? $A \preceq B \land B \preceq A \Rightarrow A \approx B$
Back: The Schröder-Bernstein theorem.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
The following is a visual depiction of what theorem?
!
Back: The Schröder-Bernstein theorem.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider injections f \colon A \rightarrow B and g \colon B \rightarrow A. What set is "reflected" in the proof of the Schröder-Bernstein theorem?
Back: A - \mathop{\text{ran}}g
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic The proof of the Schröder-Bernstein theorem uses concepts from what "paradox"? Back: Hilbert's paradox of the Grand Hotel. Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
END%%
%%ANKI
Basic
Consider this visual proof of the Schröder-Bernstein theorem. The first yellow segment corresponds to what set?
!
Back: A - \mathop{\text{ran}}g
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider this visual proof of the Schröder-Bernstein theorem. The second yellow segment corresponds to what set?
!
Back: g[\![f[\![A - \mathop{\text{ran}}g]\!]]\!]
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Consider injections f \colon A \rightarrow B and g \colon B \rightarrow A. Then h \colon A \rightarrow B is a bijection where:
- {
C_0}={A - \mathop{\text{ran} }g} and {C_{n^+}}={g[\![f[\![C_n]\!]]\!]}; h(x) ={f(x)} if {x \in \bigcup_{n} C_n};h(x) ={g^{-1}(x)} otherwise. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Hilbert's Hotel
Consider a hypothetical hotel with rooms numbered 1, 2, 3, and so on with no upper limit. That is, there is a countably infinite number of rooms in this hotel. Furthermore, it's assumed every room is occupied.
Hilbert's hotel shows that any finite or countably infinite number of additional guests can still be accommodated for.
%%ANKI Basic How many rooms exist in Hilbert's Hotel? Back: A countably infinite number. Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
END%%
%%ANKI Basic What does Hilbert's Hotel assume about every one of its rooms? Back: That they are occupied. Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
END%%
%%ANKI
Basic
How many rooms are there assumed to be in Hilbert's Hotel?
Back: A countably infinite number of them, i.e. \omega.
Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
END%%
%%ANKI
Basic
Add one guest to Hilbert's Hotel. Typically, the occupant of room n moves to what room?
Back: n + 1
Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
END%%
%%ANKI
Basic
Add k \in \mathbb{N} guests to Hilbert's Hotel. Typically, the occupant of room n moves to what room?
Back: n + k
Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
END%%
%%ANKI
Basic
Add a countably infinite number of guests to Hilbert's Hotel. Typically, the occupant of room n moves to what room?
Back: 2n
Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
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Basic
Add a countably infinite number of guests to Hilbert's Hotel. Moving occupant of room n to room 2n makes which rooms available?
Back: All odd-numbered rooms.
Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
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%%ANKI Basic What "paradox" does Hilbert's Hotel raise? Back: A fully occupied hotel can still make room for more guests. Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
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Basic
Hilbert's paradox of the Grand Hotel illustates the existence of what mathematical entity?
Back: A bijection between any countably infinite set and \mathbb{N}.
Reference: “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.
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Bibliography
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
- “Hilbert’s Paradox of the Grand Hotel.” In Wikipedia, December 23, 2024. https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel.