r
/
notebook
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Additional notes on cardinal numbers.

main
Joshua Potter 2024-12-16 11:37:31 -07:00
parent 6761cc24f5
commit d10dc3d24b
3 changed files with 81 additions and 8 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

4
notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
View File

@ -966,7 +966,7 @@
"_journal/2024-11/2024-11-21.md": "951b6034d60a40dbd8201c50abf0dbb9",
"_journal/2024-11/2024-11-20.md": "951b6034d60a40dbd8201c50abf0dbb9",
"_journal/2024-11/2024-11-19.md": "d879f57154cb27cb168eb1f1f430e312",
"set/cardinality.md": "1a9a31dcc1d1bddce9a316f9a3062324",
"set/cardinality.md": "9609eca0fe79de40e43d56cd0ef45985",
"geometry/area.md": "7f947bb5ac782495a1fb4a63bb2463e7",
"_journal/2024-11-23.md": "911f82ab8aede5ecdb96493aef64b0b9",
"_journal/2024-11/2024-11-22.md": "51117030e2364dbce3a8d507dead86ae",
@ -1023,7 +1023,7 @@
"_journal/2024-12/2024-12-13.md": "d2223f90fd1ce3d82a4fbb6828a1ec56",
"_journal/2024-12/2024-12-12.md": "59e71caa4e9ebdb11a7c7549c33bed20",
"linkers/object-files.md": "77767f310330b8650a5023dd0522226c",
"_journal/2024-12-16.md": "77c27920d5b44b6315c2ea22b990eefc",
"_journal/2024-12-16.md": "6c03d49c05eb47486aae48aa40dcef9f",
"_journal/2024-12/2024-12-15.md": "be66c8808d8bb66d4e7b91db7c93c94a",
"linkers/elf.md": "83a27a7d71ca26f8f034770a31d58fa0",
"c17/strings/printf.md": "8b67cfbccaf35dd9488b73e7e5555405",

1
notes/_journal/2024-12-16.md
View File

@ -9,3 +9,4 @@ title: "2024-12-16"
- [ ] Korean (Read 1 Story)
* Notes on sections found within an [[elf|ELF]] file.
* Additional examples on cardinal numbers and their relation to the recursion theorem.

82
notes/set/cardinality.md
View File

@ -643,15 +643,15 @@ END%%
%%ANKI
Basic
Let $K$ and $L$ be sets. How is $\mathop{\text{card}}(K \cup L)$ expressed in terms of cardinal numbers?
Back: N/A. $K$ and $L$ must be *disjoint* sets for this to make sense.
Let $K$ and $L$ be sets. What does $\mathop{\text{card}}(K \cup L)$ evaluate to?
Back: N/A. $K$ and $L$ must be disjoint sets for evaluation to make sense.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1733710439139-->
END%%
%%ANKI
Basic
Let $K$ and $L$ be disjoint sets. How is $\mathop{\text{card}}(K \cup L)$ expressed in terms of cardinal numbers?
Let $K$ and $L$ be disjoint sets. What does $\mathop{\text{card}}(K \cup L)$ evaluate to?
Back: As $\kappa + \lambda$ where $\kappa = \mathop{\text{card}} K$ and $\lambda = \mathop{\text{card}} L$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1733710439142-->
@ -672,6 +672,30 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1733710439150-->
END%%
%%ANKI
Basic
How do we prove $2 + 2 = 4$ using the recursion theorem?
Back: By proving $A_2(2) = 2^{++} = 4$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219320-->
END%%
%%ANKI
Basic
How do we prove $2 + 2 = 4$ using cardinal numbers?
Back: By proving for disjoint sets $K \approx 2$ and $L \approx 2$, that $K \cup L \approx 4$ holds.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219323-->
END%%
%%ANKI
Basic
Let $m, n \in \omega$. What does $m + n$ evaluate to in terms of cardinal numbers?
Back: $\mathop{\text{card}}((m \times \{0\}) \cup (n \times \{1\}))$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219325-->
END%%
### Multiplication
Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa \cdot \lambda = \mathop{\text{card}}(K \times L)$, where $K$ and $L$ are any sets of cardinality $\kappa$ and $\lambda$, respectively.
@ -686,7 +710,7 @@ END%%
%%ANKI
Basic
Let $K$ and $L$ be sets. How is $\mathop{\text{card}}(K \times L)$ expressed in terms of cardinal numbers?
Let $K$ and $L$ be sets. What does $\mathop{\text{card}}(K \times L)$ evaluate to?
Back: As $\kappa \cdot \lambda$ where $\kappa = \mathop{\text{card}} K$ and $\lambda = \mathop{\text{card}} L$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1733710439156-->
@ -707,6 +731,30 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1733710439162-->
END%%
%%ANKI
Basic
How do we prove $2 \cdot 2 = 4$ using the recursion theorem?
Back: By proving $M_2(2) = 2 + 2 = 4$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219326-->
END%%
%%ANKI
Basic
How do we prove $2 \cdot 2 = 4$ using cardinal numbers?
Back: By proving for sets $K \approx 2$ and $L \approx 2$, that $K \times L \approx 4$ holds.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219327-->
END%%
%%ANKI
Basic
Let $m, n \in \omega$. What does $m \cdot n$ evaluate to in terms of cardinal numbers?
Back: $\mathop{\text{card}}(m \times n)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219328-->
END%%
### Exponentiation
Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa^\lambda = \mathop{\text{card}}(^LK)$, where $K$ and $L$ are any sets of cardinality $\kappa$ and $\lambda$, respectively.
@ -721,7 +769,7 @@ END%%
%%ANKI
Basic
Let $K$ and $L$ be sets. How is $\mathop{\text{card}}(^LK)$ expressed in terms of cardinal numbers?
Let $K$ and $L$ be sets. What does $\mathop{\text{card}}(^LK)$ evaluate to?
Back: As $\kappa^\lambda$ where $\kappa = \mathop{\text{card}} K$ and $\lambda = \mathop{\text{card}} L$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1733710439168-->
@ -750,6 +798,30 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1733710439174-->
END%%
%%ANKI
Basic
How do we prove $2^2 = 4$ using the recursion theorem?
Back: By proving $E_2(2) = 2 \cdot 2 = 4$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219330-->
END%%
%%ANKI
Basic
How do we prove $2^2 = 4$ using cardinal numbers?
Back: By proving for sets $K \approx 2$ and $L \approx 2$, that $^LK \approx 4$ holds.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219331-->
END%%
%%ANKI
Basic
Let $m, n \in \omega$. What does $m^n$ evaluate to in terms of cardinal numbers?
Back: $\mathop{\text{card}}(^nm)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1734374219332-->
END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).