--- 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). END%% %%ANKI Basic What does it mean for $A$ to be equinumerous to $B$? Back: There exists a bijection between $A$ and $B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $A \approx B$. Then what must exist? Back: A bijection between $A$ and $B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose there exists a one-to-one function $F$ from $A$ into $B$. When is $A \approx B$? Back: When $F$ is also onto $B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose there exists a function $F$ from $A$ onto $B$. When 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). 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). 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). END%% %%ANKI Basic What kind of argument is typically made to prove no set is equinumerous to its power set? Back: A diagonalization argument. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Who is attributed the discovery of the diagonalization argument? Back: Georg Cantor. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $g \colon A \rightarrow \mathscr{P}A$. Using a diagonalization argument, what set is *not* in $\mathop{\text{ran}}(g)$? Back: $\{ x \in A \mid x \not\in g(x) \}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $g \colon A \rightarrow \mathscr{P}A$. *Why* isn't $B = \{x \in A \mid x \not\in g(x) \}$ in $\mathop{\text{ran}}(g)$? Back: For all $x \in A$, $x \in B \Leftrightarrow x \not\in g(x)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Equivalence Concept For any sets $A$, $B$, and $C$: * $A \approx A$; * if $A \approx B$, 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). END%% %%ANKI Basic Concisely state the equivalence concept of equinumerosity in von Neumann-Bernays set theory. Back: Class $\{ \langle A, B \rangle \mid A \approx B \}$ is reflexive on the class of all sets, symmetric, and transitive. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the reflexive property of equinumerosity in FOL? Back: $\forall A, A \approx A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the symmetric property of equinumerosity in FOL? Back: $\forall A, B, A \approx B \Rightarrow B \approx A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the transitive property of equinumerosity in FOL? Back: $\forall A, B, C, A \approx B \land B \approx C \Rightarrow A \approx C$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is $\{ \langle A, B \rangle \mid A \approx B \}$ a set? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ a set? Back: Because then the field of this "relation" would be a set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation? Back: Because then the field of this "relation" would be a set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Finiteness A set is **finite** if and only if it is equinumerous to a [[natural-numbers|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). END%% %%ANKI Basic How does Enderton define an infinite set? Back: As a set not equinumerous to any natural number. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is $n \in \omega$ a finite set? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* isn't $n \in \omega$ a finite set? Back: N/A. It is. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is $\omega$ a finite set? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* isn't $\omega$ a finite set? Back: There is no natural number equinumerous to $\omega$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Pigeonhole Principle No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself. Likewise, any set equinumerous to a proper subset of itself must be infinite. %%ANKI Basic How does Enderton state the pigeonhole principle for $\omega$? Back: No natural number is equinumerous to a proper subset of itself. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does Enderton state the pigeonhole principle for finite sets? Back: No finite set is equinumerous to a proper subset of itself. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m \in n \in \omega$. What principle precludes $m \approx n$? Back: The pigeonhole principle. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a set and $n \in \omega$ such that $S \approx n$. For $m \in \omega$, when might $S \approx m$? Back: *Only* if $m = n$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the generalization of the pigeonhole principle for $\omega$? Back: The pigeonhole principle for finite sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of the pigeonhole principle for finite sets? Back: The pigeonhole principle for $\omega$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What name is given to the following theorem? $$\text{No finite set is equinumerous to a proper subset of itself.}$$ Back: The pigeonhole principle. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. Is $f$ a bijection? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. *Why* must $f$ be surjective? Back: Otherwise $f$ is a bijection between $S$ and a proper subset of $S$, a contradiction. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. Is $f$ a bijection? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. *Why* must $f$ be injective? Back: Otherwise $f$ is a bijection between a proper subset of $S$ and $S$, a contradiction. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does the contrapositive of the pigeonhole principle state? Back: Any set equinumerous to a proper subset of itself is infinite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What general strategy is used to prove $\omega$ is an infinite set? Back: Prove $\omega$ is equinumerous to a proper subset of itself. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Cardinal Numbers A **cardinal number** is a set that is $\mathop{\text{card}} A$ for some set $A$. The set $\mathop{\text{card}} A$ is defined such that * For any sets $A$ and $B$, $\mathop{\text{card}}A = \mathop{\text{card}}B$ iff $A \approx B$. * For a finite set $A$, $\mathop{\text{card}}A$ is the natural number $n$ for which $A \approx n$. %%ANKI Basic How is the cardinal number of set $A$ denoted? Back: As $\mathop{\text{card}} A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $A$ is finite. What does $\mathop{\text{card}} A$ evaluate to? Back: The unique $n \in \omega$ such that $A \approx n$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider $n \in \omega$. What does $\mathop{\text{card}} n$ evaluate to? Back: $n$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $a$, $b$, and $c$ are distinct objects. What does $\mathop{\text{card}} \{a, b, c\}$ evaluate to? Back: $3$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does Enderton refer to by the "process called 'counting'"? Back: Choosing a one-to-one correspondence between two sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze A {cardinal number} is a set that is {$\mathop{\text{card}} A$} for some set $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How do cardinal numbers relate to equinumerosity? Back: For any sets $A$ and $B$, $\mathop{\text{card}} A = \mathop{\text{card}} B$ iff $A \approx B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic According to Enderton, what is the "essential demand" for defining numbers? Back: Defining cardinal numbers such that for any sets $A$ and $B$, $\mathop{\text{card}} A = \mathop{\text{card}} B$ iff $A \approx B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What name is given to $\mathop{\text{card}} \omega$? Back: $\aleph_0$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Who is attributed the assignment $\mathop{\text{card}} \omega = \aleph_0$? Back: Georg Cantor. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Bibliography * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).