1693 lines
59 KiB
Markdown
1693 lines
59 KiB
Markdown
---
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title: Cardinality
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set::cardinality
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tags:
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- set
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---
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## Equinumerosity
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We say set $A$ is **equinumerous** to set $B$, written ($A \approx B$) if and only if there exists a [[set/functions#Injections|one-to-one]] function from $A$ [[set/functions#Surjections|onto]] $B$.
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%%ANKI
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Basic
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Suppose $A$ is equinumerous to $B$. How does Enderton denote this?
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Back: $A \approx B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060344-->
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END%%
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%%ANKI
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Basic
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What does it mean for $A$ to be equinumerous to $B$?
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Back: There exists a bijection between $A$ and $B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060352-->
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END%%
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%%ANKI
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Basic
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Suppose $A \approx B$. Then what must exist?
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Back: A bijection between $A$ and $B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060355-->
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END%%
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%%ANKI
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Basic
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Suppose there exists a one-to-one function $F$ from $A$ into $B$. When does this imply $A \approx B$?
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Back: When $F$ is also onto $B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060358-->
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END%%
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%%ANKI
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Basic
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Suppose there exists a function $F$ from $A$ onto $B$. When does this imply $A \approx B$?
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Back: When $F$ is also one-to-one.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060362-->
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END%%
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%%ANKI
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Basic
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Suppose there exists a one-to-one function $F$ from $A$ onto $B$. When does this imply $A \approx B$?
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Back: Always, by definition.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060366-->
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END%%
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### Power Sets
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No set is equinumerous to its [[set/index#Power Set Axiom|power set]]. This is typically shown using a diagonalization argument.
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For any set $A$, $^A2 \approx \mathscr{P}(A)$.
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%%ANKI
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Basic
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What kind of argument is typically used to show $A \not\approx {\mathscr{P}(A)}$?
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Back: A diagonalization argument.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309208-->
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END%%
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%%ANKI
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Basic
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Who is attributed the discovery of the diagonalization argument?
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Back: Georg Cantor.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309212-->
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END%%
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%%ANKI
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Basic
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Let $g \colon A \rightarrow \mathscr{P}A$. Using a diagonalization argument, what set is *not* in $\mathop{\text{ran}}(g)$?
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Back: $\{ x \in A \mid x \not\in g(x) \}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309216-->
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END%%
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%%ANKI
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Basic
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Let $g \colon A \rightarrow \mathscr{P}A$. *Why* isn't $B = \{x \in A \mid x \not\in g(x) \}$ in $\mathop{\text{ran}}(g)$?
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Back: For all $x \in A$, $x \in B \Leftrightarrow x \not\in g(x)$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309221-->
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END%%
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%%ANKI
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Basic
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For any set $A$, $\mathscr{P}(A)$ is equinumerous to what set of functions?
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Back: $^A2$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734802664285-->
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END%%
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%%ANKI
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Basic
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Let $A$ be any set. How is bijection $H \colon \,^A2 \rightarrow \mathscr{P}(A)$ typically defined?
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Back: $H(f) = \{a \in A \mid f(a) = 1\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734802664293-->
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END%%
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%%ANKI
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Basic
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For any set $A$, $^A2$ is equinumerous to what of $A$?
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Back: Its powerset.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734802664297-->
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END%%
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%%ANKI
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Basic
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What kind of argument is typically used to show $A \not\approx {^A2}$?
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Back: A diagonalization argument.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734802664301-->
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END%%
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%%ANKI
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Basic
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For any set $A$, what is the cardinality of its powerset?
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Back: $2^{\mathop{\text{card}}A}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734803273736-->
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END%%
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%%ANKI
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Basic
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What does $\mathop{\text{card}} \mathscr{P}(A)$ evaluate to?
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Back: $2^{\mathop{\text{card}}A}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734803273739-->
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END%%
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%%ANKI
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Basic
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The cardinality of what set equals $2^{\mathop{\text{card}}A}$?
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Back: $\mathscr{P}(A)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734803273741-->
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END%%
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%%ANKI
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Basic
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What does $\mathop{\text{card}} \mathscr{P}(\omega)$ evaluate to?
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Back: $2^{\aleph_0}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734803273742-->
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END%%
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%%ANKI
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Basic
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Why is the "power set" named the way it is?
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Back: Because $\mathop{\text{card}} \mathscr{P}(A)$ equals $2$ to the power of $\mathop{\text{card}} A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734803273743-->
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END%%
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%%ANKI
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Basic
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*How* do we know $\aleph_0 \not\approx 2^{\aleph_0}$ holds?
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Back: No set is equinumerous to its power set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734803273744-->
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END%%
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%%ANKI
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Basic
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For any cardinal number $\kappa$, *how* do we know $\kappa \not\approx 2^\kappa$?
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Back: No set is equinumerous to its power set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1734803273745-->
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END%%
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### Equivalence Concept
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For any sets $A$, $B$, and $C$:
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* $A \approx A$;
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* if $A \approx B$, then $B \approx A$;
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* if $A \approx B$ and $B \approx C$, then $A \approx C$.
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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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%%ANKI
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Basic
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Concisely state the equivalence concept of equinumerosity in Zermelo-Fraenkel set theory.
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Back: For all sets $A$, $B$, and $C$:
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* $A \approx A$;
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* $A \approx B \Rightarrow B \approx A$;
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* $A \approx B \land B \approx C \Rightarrow A \approx C$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060370-->
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END%%
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%%ANKI
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Basic
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Concisely state the equivalence concept of equinumerosity in von Neumann-Bernays set theory.
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Back: Class $\{ \langle A, B \rangle \mid A \approx B \}$ is reflexive on the class of all sets, symmetric, and transitive.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060374-->
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END%%
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%%ANKI
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Basic
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What is the reflexive property of equinumerosity in FOL?
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Back: $\forall A, A \approx A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060379-->
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END%%
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%%ANKI
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Basic
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What is the symmetric property of equinumerosity in FOL?
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Back: $\forall A, B, A \approx B \Rightarrow B \approx A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060383-->
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END%%
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%%ANKI
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Basic
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What is the transitive property of equinumerosity in FOL?
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Back: $\forall A, B, C, A \approx B \land B \approx C \Rightarrow A \approx C$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060387-->
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END%%
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%%ANKI
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Basic
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Is $\{ \langle A, B \rangle \mid A \approx B \}$ a set?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060390-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ a set?
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Back: Because then the field of this "relation" would be a set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060394-->
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END%%
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%%ANKI
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Basic
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Is $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060398-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation?
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Back: Because then the field of this "relation" would be a set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060403-->
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END%%
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## Finiteness
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A set is **finite** if and only if it is equinumerous to a [[natural-numbers|natural number]]. Otherwise it is **infinite**.
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%%ANKI
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Basic
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How does Enderton define a finite set?
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Back: As a set equinumerous to some natural number.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231320-->
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END%%
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%%ANKI
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Basic
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How does Enderton define an infinite set?
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Back: As a set not equinumerous to any natural number.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231330-->
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END%%
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%%ANKI
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Basic
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Is $n \in \omega$ a finite set?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231336-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $n \in \omega$ a finite set?
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Back: N/A. It is.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231342-->
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END%%
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%%ANKI
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Basic
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Is $\omega$ a finite set?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231347-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $\omega$ a finite set?
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Back: There is no natural number equinumerous to $\omega$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231353-->
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END%%
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### Pigeonhole Principle
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No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself.
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Likewise, any set equinumerous to a proper subset of itself must be infinite.
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%%ANKI
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Basic
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How does Enderton state the pigeonhole principle for $\omega$?
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Back: No natural number is equinumerous to a proper subset of itself.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231358-->
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END%%
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%%ANKI
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Basic
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How does Enderton state the pigeonhole principle for finite sets?
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Back: No finite set is equinumerous to a proper subset of itself.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231364-->
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END%%
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%%ANKI
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Basic
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Let $m \in n \in \omega$. What principle precludes $m \approx n$?
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Back: The pigeonhole principle.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231369-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a set and $n \in \omega$ such that $S \approx n$. For $m \in \omega$, when might $S \approx m$?
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Back: *Only* if $m = n$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231374-->
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END%%
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%%ANKI
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Basic
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What is the generalization of the pigeonhole principle for $\omega$?
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Back: The pigeonhole principle for finite sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231379-->
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END%%
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%%ANKI
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Basic
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What is the specialization of the pigeonhole principle for finite sets?
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Back: The pigeonhole principle for $\omega$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231385-->
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END%%
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%%ANKI
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Basic
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What name is given to the following theorem? $$\text{No finite set is equinumerous to a proper subset of itself.}$$
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Back: The pigeonhole principle.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231391-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. Is $f$ a bijection?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231396-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. *Why* must $f$ be surjective?
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Back: Otherwise $f$ is a bijection between $S$ and a proper subset of $S$, a contradiction.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231401-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. Is $f$ a bijection?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231407-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. *Why* must $f$ be injective?
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Back: Otherwise $f$ is a bijection between a proper subset of $S$ and $S$, a contradiction.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231412-->
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END%%
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%%ANKI
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Basic
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What does the contrapositive of the pigeonhole principle state?
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Back: Any set equinumerous to a proper subset of itself is infinite.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760079-->
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END%%
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%%ANKI
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Basic
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What general strategy is used to prove $\omega$ is an infinite set?
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Back: Prove $\omega$ is equinumerous to a proper subset of itself.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760085-->
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END%%
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## Cardinal Numbers
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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
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* For any sets $A$ and $B$, $\mathop{\text{card}}A = \mathop{\text{card}}B$ iff $A \approx B$.
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* For a finite set $A$, $\mathop{\text{card}}A$ is the natural number $n$ for which $A \approx n$.
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%%ANKI
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Basic
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How is the cardinal number of set $A$ denoted?
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Back: As $\mathop{\text{card}} A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760088-->
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END%%
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%%ANKI
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Basic
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Suppose $A$ is finite. What does $\mathop{\text{card}} A$ evaluate to?
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Back: The unique $n \in \omega$ such that $A \approx n$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760091-->
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END%%
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%%ANKI
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Basic
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Consider $n \in \omega$. What does $\mathop{\text{card}} n$ evaluate to?
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Back: $n$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760094-->
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END%%
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%%ANKI
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Basic
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Suppose $a$, $b$, and $c$ are distinct objects. What does $\mathop{\text{card}} \{a, b, c\}$ evaluate to?
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Back: $3$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760097-->
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END%%
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%%ANKI
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Basic
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What does Enderton refer to by the "process called 'counting'"?
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Back: Choosing a one-to-one correspondence between two sets.
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1733407760099-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
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).
|
||
<!--ID: 1733407760102-->
|
||
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).
|
||
<!--ID: 1733407760105-->
|
||
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).
|
||
<!--ID: 1733407760108-->
|
||
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).
|
||
<!--ID: 1733407760110-->
|
||
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).
|
||
<!--ID: 1733407760113-->
|
||
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).
|
||
<!--ID: 1733675315442-->
|
||
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).
|
||
<!--ID: 1733675315447-->
|
||
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).
|
||
<!--ID: 1733675315450-->
|
||
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).
|
||
<!--ID: 1733675315453-->
|
||
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).
|
||
<!--ID: 1733675315456-->
|
||
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).
|
||
<!--ID: 1733675315459-->
|
||
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).
|
||
<!--ID: 1733675315461-->
|
||
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).
|
||
<!--ID: 1733675315464-->
|
||
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).
|
||
<!--ID: 1733675315466-->
|
||
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).
|
||
<!--ID: 1733675315469-->
|
||
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).
|
||
<!--ID: 1733675315471-->
|
||
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).
|
||
<!--ID: 1733675315474-->
|
||
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).
|
||
<!--ID: 1733675315477-->
|
||
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).
|
||
<!--ID: 1733675315479-->
|
||
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).
|
||
<!--ID: 1733675522739-->
|
||
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).
|
||
<!--ID: 1733693785274-->
|
||
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).
|
||
<!--ID: 1733693785281-->
|
||
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).
|
||
<!--ID: 1733693785284-->
|
||
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).
|
||
<!--ID: 1733693785287-->
|
||
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).
|
||
<!--ID: 1733675522748-->
|
||
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).
|
||
<!--ID: 1733693785290-->
|
||
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).
|
||
<!--ID: 1733693785292-->
|
||
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).
|
||
<!--ID: 1733675522751-->
|
||
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).
|
||
<!--ID: 1735074143693-->
|
||
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).
|
||
<!--ID: 1733710439132-->
|
||
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).
|
||
<!--ID: 1733710439142-->
|
||
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).
|
||
<!--ID: 1733710439146-->
|
||
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).
|
||
<!--ID: 1733710439150-->
|
||
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).
|
||
<!--ID: 1734374219320-->
|
||
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).
|
||
<!--ID: 1734374219323-->
|
||
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).
|
||
<!--ID: 1734374219325-->
|
||
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).
|
||
<!--ID: 1734520487310-->
|
||
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).
|
||
<!--ID: 1734520487313-->
|
||
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).
|
||
<!--ID: 1734520487316-->
|
||
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).
|
||
<!--ID: 1734520487319-->
|
||
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).
|
||
<!--ID: 1734520487326-->
|
||
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).
|
||
<!--ID: 1734520487330-->
|
||
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).
|
||
<!--ID: 1734520487333-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
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).
|
||
<!--ID: 1735074143694-->
|
||
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).
|
||
<!--ID: 1735074143695-->
|
||
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).
|
||
<!--ID: 1735074143696-->
|
||
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).
|
||
<!--ID: 1735074143697-->
|
||
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).
|
||
<!--ID: 1733710439153-->
|
||
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).
|
||
<!--ID: 1733710439156-->
|
||
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).
|
||
<!--ID: 1733710439159-->
|
||
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).
|
||
<!--ID: 1733710439162-->
|
||
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).
|
||
<!--ID: 1733693785295-->
|
||
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).
|
||
<!--ID: 1734374219326-->
|
||
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).
|
||
<!--ID: 1734374219327-->
|
||
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).
|
||
<!--ID: 1734374219328-->
|
||
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).
|
||
<!--ID: 1734520487336-->
|
||
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).
|
||
<!--ID: 1734520487339-->
|
||
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).
|
||
<!--ID: 1734520487342-->
|
||
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).
|
||
<!--ID: 1734520487345-->
|
||
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).
|
||
<!--ID: 1734520487349-->
|
||
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).
|
||
<!--ID: 1734520487352-->
|
||
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).
|
||
<!--ID: 1734520487356-->
|
||
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).
|
||
<!--ID: 1734520487359-->
|
||
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).
|
||
<!--ID: 1734803273746-->
|
||
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).
|
||
<!--ID: 1735074143698-->
|
||
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).
|
||
<!--ID: 1735074143699-->
|
||
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).
|
||
<!--ID: 1735074143700-->
|
||
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).
|
||
<!--ID: 1735074143701-->
|
||
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).
|
||
<!--ID: 1735074143702-->
|
||
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).
|
||
<!--ID: 1735074143703-->
|
||
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).
|
||
<!--ID: 1735074143704-->
|
||
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).
|
||
<!--ID: 1733710439165-->
|
||
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).
|
||
<!--ID: 1733710439168-->
|
||
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).
|
||
<!--ID: 1733710755119-->
|
||
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).
|
||
<!--ID: 1733710439171-->
|
||
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).
|
||
<!--ID: 1733710439174-->
|
||
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).
|
||
<!--ID: 1734374219330-->
|
||
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).
|
||
<!--ID: 1734374219331-->
|
||
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).
|
||
<!--ID: 1734374219332-->
|
||
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).
|
||
<!--ID: 1734520487363-->
|
||
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).
|
||
<!--ID: 1734520487368-->
|
||
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).
|
||
<!--ID: 1734520487372-->
|
||
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).
|
||
<!--ID: 1734520487376-->
|
||
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).
|
||
<!--ID: 1734520487381-->
|
||
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).
|
||
<!--ID: 1734520487384-->
|
||
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).
|
||
<!--ID: 1734520487388-->
|
||
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).
|
||
<!--ID: 1735074143705-->
|
||
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).
|
||
<!--ID: 1735074143706-->
|
||
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).
|
||
<!--ID: 1735074143707-->
|
||
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).
|
||
<!--ID: 1735074143708-->
|
||
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).
|
||
<!--ID: 1735074143709-->
|
||
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).
|
||
<!--ID: 1735074143710-->
|
||
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).
|
||
<!--ID: 1735074143711-->
|
||
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).
|
||
<!--ID: 1735074143712-->
|
||
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).
|
||
<!--ID: 1735353438914-->
|
||
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).
|
||
<!--ID: 1735493305902-->
|
||
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).
|
||
<!--ID: 1735353438921-->
|
||
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).
|
||
<!--ID: 1735493305907-->
|
||
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).
|
||
<!--ID: 1735353438924-->
|
||
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).
|
||
<!--ID: 1735493305910-->
|
||
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).
|
||
<!--ID: 1735493305914-->
|
||
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).
|
||
<!--ID: 1735493305917-->
|
||
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).
|
||
<!--ID: 1735353438928-->
|
||
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).
|
||
<!--ID: 1735493305921-->
|
||
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).
|
||
<!--ID: 1735353438932-->
|
||
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).
|
||
<!--ID: 1735493305924-->
|
||
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).
|
||
<!--ID: 1735493305927-->
|
||
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).
|
||
<!--ID: 1735493305931-->
|
||
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).
|
||
<!--ID: 1735493305934-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
{$<$} on cardinal numbers corresponds to {$\prec$} on sets.
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1735493305942-->
|
||
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).
|
||
<!--ID: 1735493305947-->
|
||
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).
|
||
<!--ID: 1735493305952-->
|
||
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).
|
||
<!--ID: 1735493305957-->
|
||
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).
|
||
<!--ID: 1735493305962-->
|
||
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).
|
||
<!--ID: 1735493767848-->
|
||
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).
|
||
<!--ID: 1735493767854-->
|
||
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).
|
||
<!--ID: 1735522988804-->
|
||
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).
|
||
<!--ID: 1735522988810-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the smallest cardinal number?
|
||
Back: $0$
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1735522988813-->
|
||
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).
|
||
<!--ID: 1735522988817-->
|
||
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).
|
||
<!--ID: 1736702905232-->
|
||
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).
|
||
<!--ID: 1736702905238-->
|
||
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).
|
||
<!--ID: 1736702905241-->
|
||
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).
|
||
<!--ID: 1736702905244-->
|
||
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).
|
||
<!--ID: 1736702905248-->
|
||
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).
|
||
<!--ID: 1736702905252-->
|
||
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).
|
||
<!--ID: 1736702905257-->
|
||
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).
|
||
<!--ID: 1736702905262-->
|
||
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).
|
||
<!--ID: 1736702905267-->
|
||
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).
|
||
<!--ID: 1736711693540-->
|
||
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).
|
||
<!--ID: 1736711693542-->
|
||
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).
|
||
<!--ID: 1736711693543-->
|
||
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).
|
||
<!--ID: 1736711693544-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
The following is a visual depiction of what theorem?
|
||
![[schroder-bernstein.png]]
|
||
Back: The Schröder-Bernstein theorem.
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1736711693545-->
|
||
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).
|
||
<!--ID: 1736711693546-->
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693547-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Consider this visual proof of the Schröder-Bernstein theorem. The first yellow segment corresponds to what set?
|
||
![[schroder-bernstein.png]]
|
||
Back: $A - \mathop{\text{ran}}g$
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1736711693548-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Consider this visual proof of the Schröder-Bernstein theorem. The second yellow segment corresponds to what set?
|
||
![[schroder-bernstein.png]]
|
||
Back: $g[\![f[\![A - \mathop{\text{ran}}g]\!]]\!]$
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1736711693549-->
|
||
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).
|
||
<!--ID: 1736711693550-->
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693551-->
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693552-->
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693553-->
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693554-->
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693555-->
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693556-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693557-->
|
||
END%%
|
||
|
||
%%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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693558-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516).
|
||
<!--ID: 1736711693559-->
|
||
END%%
|
||
|
||
## 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](https://en.wikipedia.org/w/index.php?title=Hilbert%27s_paradox_of_the_Grand_Hotel&oldid=1264833516). |