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"set/cardinality.md": "02191c855899826b9e379adfa1e9b98e", + "set/cardinality.md": "a684ba2638e90d6d5647054230b3f35f", "geometry/area.md": "7f947bb5ac782495a1fb4a63bb2463e7", "_journal/2024-11-23.md": "911f82ab8aede5ecdb96493aef64b0b9", "_journal/2024-11/2024-11-22.md": "51117030e2364dbce3a8d507dead86ae", diff --git a/notes/_journal/2025-01-12.md b/notes/_journal/2025-01-12.md index 215bc84..807fdac 100644 --- a/notes/_journal/2025-01-12.md +++ b/notes/_journal/2025-01-12.md @@ -6,4 +6,6 @@ title: "2025-01-12" - [x] KoL - [x] OGS - [ ] Sheet Music (10 min.) -- [ ] Korean (Read 1 Story) \ No newline at end of file +- [ ] Korean (Read 1 Story) + +* Notes on the [[cardinality#Schröder-Bernstein Theorem|Schröder-Bernstein Theorem]]. \ No newline at end of file diff --git a/notes/set/cardinality.md b/notes/set/cardinality.md index e3c3d14..f5654e3 100644 --- a/notes/set/cardinality.md +++ b/notes/set/cardinality.md @@ -1236,7 +1236,7 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% -### Ordering +## 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 $$\mathop{\text{card}}A \leq \mathop{\text{card}}B \text{ if and only if } A \preceq B.$$ @@ -1520,6 +1520,174 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +END%% + ## Bibliography -* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). \ No newline at end of file +* 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). \ No newline at end of file diff --git a/notes/set/images/schroder-bernstein.png b/notes/set/images/schroder-bernstein.png new file mode 100644 index 0000000..3b8f215 Binary files /dev/null and b/notes/set/images/schroder-bernstein.png differ