148 lines
4.5 KiB
Markdown
148 lines
4.5 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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## Overview
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We say set $A$ is **equinumerous** to set $B$, written ($A \approx B$) if and only if there exists a [[functions#Injections|one-to-one]] function from $A$ [[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 is $A \approx B$?
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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 is $A \approx B$?
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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 is $A \approx B$?
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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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## 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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## Bibliography
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* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |