notebook/notes/set/cardinality.md

---
title: Cardinality
TARGET DECK: Obsidian::STEM
FILE TAGS: set::cardinality
tags:
  - set
---

## Equinumerosity

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]] $B$.

%%ANKI
Basic
Suppose $A$ is equinumerous to $B$. How does Enderton denote this?
Back: $A \approx B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
What does it mean for $A$ to be equinumerous to $B$?
Back: There exists a bijection between $A$ and $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732295060352-->
END%%

%%ANKI
Basic
Suppose $A \approx B$. Then what must exist?
Back: A bijection between $A$ and $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732295060355-->
END%%

%%ANKI
Basic
Suppose there exists a one-to-one function $F$ from $A$ into $B$. When is $A \approx B$?
Back: When $F$ is also onto $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Suppose there exists a function $F$ from $A$ onto $B$. When is $A \approx B$?
Back: When $F$ is also one-to-one.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Suppose there exists a one-to-one function $F$ from $A$ onto $B$. When is $A \approx B$?
Back: Always, by definition.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

### Power Sets

No set is equinumerous to its [[set/index#Power Set Axiom|power set]]. This is typically shown using a diagonalization argument.

%%ANKI
Basic
What basic set operation produces a new set the original isn't equinumerous to?
Back: The power set operation.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
What kind of argument is typically made to prove no set is equinumerous to its power set?
Back: A diagonalization argument.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Who is attributed the discovery of the diagonalization argument?
Back: Georg Cantor.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Let $g \colon A \rightarrow \mathscr{P}A$. Using a diagonalization argument, what set is *not* in $\mathop{\text{ran}}(g)$?
Back: $\{ x \in A \mid x \not\in g(x) \}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Let $g \colon A \rightarrow \mathscr{P}A$. *Why* isn't $B = \{x \in A \mid x \not\in g(x) \}$ in $\mathop{\text{ran}}(g)$?
Back: For all $x \in A$, $x \in B \Leftrightarrow x \not\in g(x)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

### Equivalence Concept

For any sets $A$, $B$, and $C$:

* $A \approx A$;
* if $A \approx B$, then $B \approx A$;
* if $A \approx B$ and $B \approx C$, then $A \approx C$.

Notice though that $\{ \langle A, B \rangle \mid A \approx B \}$ is *not* an equivalence relation since the equivalence concept of equinumerosity concerns *all* sets.

%%ANKI
Basic
Concisely state the equivalence concept of equinumerosity in Zermelo-Fraenkel set theory.
Back: For all sets $A$, $B$, and $C$:
* $A \approx A$;
* $A \approx B \Rightarrow B \approx A$;
* $A \approx B \land B \approx C \Rightarrow A \approx C$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732295060370-->
END%%

%%ANKI
Basic
Concisely state the equivalence concept of equinumerosity in von Neumann-Bernays set theory.
Back: Class $\{ \langle A, B \rangle \mid A \approx B \}$ is reflexive on the class of all sets, symmetric, and transitive.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
What is the reflexive property of equinumerosity in FOL?
Back: $\forall A, A \approx A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
What is the symmetric property of equinumerosity in FOL?
Back: $\forall A, B, A \approx B \Rightarrow B \approx A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
What is the transitive property of equinumerosity in FOL?
Back: $\forall A, B, C, A \approx B \land B \approx C \Rightarrow A \approx C$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732295060387-->
END%%

%%ANKI
Basic
Is $\{ \langle A, B \rangle \mid A \approx B \}$ a set?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732295060390-->
END%%

%%ANKI
Basic
*Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ a set?
Back: Because then the field of this "relation" would be a set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732295060394-->
END%%

%%ANKI
Basic
Is $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732295060398-->
END%%

%%ANKI
Basic
*Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation?
Back: Because then the field of this "relation" would be a set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732295060403-->
END%%

## Finiteness

A set is **finite** if and only if it is equinumerous to some [[natural-numbers|natural number]]. Otherwise it is **infinite**.

%%ANKI
Basic
How does Enderton define a finite set?
Back: As a set equinumerous to some natural number.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
How does Enderton define an infinite set?
Back: As a set not equinumerous to any natural number.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Is $n \in \omega$ a finite set?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231336-->
END%%

%%ANKI
Basic
*Why* isn't $n \in \omega$ a finite set?
Back: N/A. It is.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231342-->
END%%

%%ANKI
Basic
Is $\omega$ a finite set?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231347-->
END%%

%%ANKI
Basic
*Why* isn't $\omega$ a finite set?
Back: There is no natural number equinumerous to $\omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231353-->
END%%

### Pigeonhole Principle

No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself.

%%ANKI
Basic
How does Enderton state the pigeonhole principle for $\omega$?
Back: No natural number is equinumerous to a proper subset of itself.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231358-->
END%%

%%ANKI
Basic
How does Enderton state the pigeonhole principle for finite sets?
Back: No finite set is equinumerous to a proper subset of itself.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231364-->
END%%

%%ANKI
Basic
Let $m \in n \in \omega$. What principle precludes $m \approx n$?
Back: The pigeonhole principle.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231369-->
END%%

%%ANKI
Basic
Let $S$ be a set and $n \in \omega$ such that $S \approx n$. For $m \in \omega$, when might $S \approx m$?
Back: *Only* if $m = n$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231374-->
END%%

%%ANKI
Basic
What is the generalization of the pigeonhole principle for $\omega$?
Back: The pigeonhole principle for finite sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231379-->
END%%

%%ANKI
Basic
What is the specialization of the pigeonhole principle for finite sets?
Back: The pigeonhole principle for $\omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231385-->
END%%

%%ANKI
Basic
What name is given to the following theorem? $$\text{No finite set is equinumerous to a proper subset of itself.}$$
Back: The pigeonhole principle.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231391-->
END%%

%%ANKI
Basic
Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. Is $f$ a bijection?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231396-->
END%%

%%ANKI
Basic
Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. *Why* must $f$ be surjective?
Back: Otherwise $f$ is a bijection between $S$ and a proper subset of $S$, a contradiction.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231401-->
END%%

%%ANKI
Basic
Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. Is $f$ a bijection?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231407-->
END%%

%%ANKI
Basic
Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. *Why* must $f$ be injective?
Back: Otherwise $f$ is a bijection between a proper subset of $S$ and $S$, a contradiction.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1732545231412-->
END%%

## Bibliography

* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).