notebook/notes/set/cardinality.md

title	TARGET DECK	FILE TAGS	tags
Cardinality	Obsidian::STEM	set::cardinality	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 function from A set/functions#Surjections 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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%%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).

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%%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).

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%%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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%%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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%%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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Power Sets

No set is equinumerous to its set/index#Power Set Axiom. 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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%%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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%%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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%%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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%%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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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).

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%%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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%%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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%%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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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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Finiteness

A set is finite if and only if it is equinumerous to a natural-numbers. Otherwise it is infinite.

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

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%%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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%%ANKI Basic Is n \in \omega a finite set? Back: Yes. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%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).

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%%ANKI Basic Is \omega a finite set? Back: No. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%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).

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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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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Bibliography

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