diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 88da64d..df2464a 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -229,7 +229,7 @@ "_journal/2024-02/2024-02-14.md": "aa009f9569e175a8104b0537ebcc5520", "_journal/2024-02-16.md": "5cc129254afd553829be3364facd23db", "_journal/2024-02/2024-02-15.md": "16cb7563d404cb543719b7bb5037aeed", - "algebra/floor-ceiling.md": "aa89a485faeb7e71ef68da9200544184", + "algebra/floor-ceiling.md": "a22efe853ad1234b2d3e0d7cc7e6fc47", "algebra/index.md": "90b842eb694938d87c7c68779a5cacd1", "algorithms/binary-search.md": "8533a05ea372e007ab4e8a36fd2772a9", "_journal/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048", @@ -460,7 +460,7 @@ "_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c", "_journal/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", "hashing/direct-addressing.md": "f75cc22e74ae974fe4f568a2ee9f951f", - "hashing/index.md": "378d86af6fe91fa93a6d9a6d8e52a1bf", + "hashing/index.md": "e90b8e0eb7b93fda713c364027da71c2", "set/classes.md": "6776b4dc415021e0ef60b323b5c2d436", "_journal/2024-05-19.md": "fddd90fae08fab9bd83b0ef5d362c93a", "_journal/2024-05/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", @@ -483,7 +483,7 @@ "_journal/2024-05/2024-05-25.md": "3e8a0061fa58a6e5c48d12800d1ab869", "_journal/2024-05-27.md": "b36636d10eab34380f17f288868df3ae", "_journal/2024-05/2024-05-26.md": "abe84b5beae74baa25501c818e64fc95", - "algebra/set.md": "237c9817db3d26f247e81ceb6a86b97f", + "algebra/set.md": "204dba5e6da6257c01440758c17d305c", "algebra/boolean.md": "ee41e624f4d3d3aca00020d9a9ae42c8", "git/merge-conflicts.md": "761ad6137ec51d3877f7d5b3615ca5cb", "_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337", @@ -537,7 +537,7 @@ "set/functions.md": "4fd3388fb21c77e96c6cfb703f3ed153", "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4", "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307", - "lambda-calculus/beta-reduction.md": "5386403713b42ee1831d15c84de133ba", + "lambda-calculus/beta-reduction.md": "e233c8352a8180d19f7b717946c379d1", "_journal/2024-06-16.md": "ded6ab660ecc7c3dce3afd2e88e5a725", "_journal/2024-06/2024-06-15.md": "c3a55549da9dfc2770bfcf403bf5b30b", "_journal/2024-06-17.md": "63df6757bb3384e45093bf2b9456ffac", @@ -589,7 +589,10 @@ "_journal/2024-07-09.md": "00c357e9cfac6de17825b02fdbd00c80", "_journal/2024-07/2024-07-08.md": "03ed5604e680ac9742ee99ae4b1eee8b", "_journal/2024-07-10.md": "2bb3db1f506f4ec7726cb5f2ed2daf24", - "_journal/2024-07/2024-07-09.md": "00c357e9cfac6de17825b02fdbd00c80" + "_journal/2024-07/2024-07-09.md": "00c357e9cfac6de17825b02fdbd00c80", + "_journal/2024-07-12.md": "247909b64d6b0dd7702d6a4482165c4d", + "_journal/2024-07/2024-07-11.md": "298cc3688675ee669b5a51d545fd61b5", + "_journal/2024-07/2024-07-10.md": "a0fe22d8be519bf435a5949999eeb4de" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-07-12.md b/notes/_journal/2024-07-12.md new file mode 100644 index 0000000..318d202 --- /dev/null +++ b/notes/_journal/2024-07-12.md @@ -0,0 +1,11 @@ +--- +title: "2024-07-12" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Notes on [[set#Index Sets|index sets]] and [[set#Function Sets|function sets]]. \ No newline at end of file diff --git a/notes/_journal/2024-07-10.md b/notes/_journal/2024-07/2024-07-10.md similarity index 100% rename from notes/_journal/2024-07-10.md rename to notes/_journal/2024-07/2024-07-10.md diff --git a/notes/_journal/2024-07/2024-07-11.md b/notes/_journal/2024-07/2024-07-11.md new file mode 100644 index 0000000..0a67ea2 --- /dev/null +++ b/notes/_journal/2024-07/2024-07-11.md @@ -0,0 +1,9 @@ +--- +title: "2024-07-11" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) \ No newline at end of file diff --git a/notes/algebra/floor-ceiling.md b/notes/algebra/floor-ceiling.md index f102ed7..48b3f13 100644 --- a/notes/algebra/floor-ceiling.md +++ b/notes/algebra/floor-ceiling.md @@ -291,7 +291,7 @@ END%% %%ANKI Basic Given `A[p:q]` and $r = \lfloor (p + q) / 2 \rfloor$, what is the size of `A[p:r]` in terms of $n = q - p + 1$? -Back: $\lceil n / 2 \rceil$. +Back: $\lceil n / 2 \rceil$ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). END%% @@ -299,7 +299,7 @@ END%% %%ANKI Basic Given `A[p:q]` and $r = \lfloor (p + q) / 2 \rfloor$, what is the size of `A[r+1:q]` in terms of $n = q - p + 1$? -Back: $\lfloor n / 2 \rfloor$. +Back: $\lfloor n / 2 \rfloor$ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). END%% diff --git a/notes/algebra/set.md b/notes/algebra/set.md index dc95996..0d9c76f 100644 --- a/notes/algebra/set.md +++ b/notes/algebra/set.md @@ -794,6 +794,304 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +## Index Sets + +Let $I$ be a set, called the **index set**. Let $F$ be a [[functions|function]] whose domain includes $I$. Then we define $$\bigcup_{i \in I} F(i) = \bigcup\,\{F(i) \mid i \in I\}$$ +and, if $I \neq \varnothing$, $$\bigcap_{i \in I} F(i) = \bigcap\, \{F(i) \mid i \in I\}$$ + +%%ANKI +Basic +What name does $I$ go by in expression $\bigcup_{i \in I} F(i)$? +Back: The "index set". +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is $\bigcup_{i \in I} F(i)$ alternatively denoted? +Back: $\bigcup_{i \in I} F_i$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What kind of mathematic object is $I$ in expression $\bigcup_{i \in I} F(i)$? +Back: A set. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What kind of mathematic object is $F$ in expression $\bigcup_{i \in I} F(i)$? +Back: A function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is $\bigcup_{i \in I} F_i$ alternatively denoted? +Back: $\bigcup_{i \in I} F(i)$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What kind of mathematic object is $F$ in expression $\bigcup_{i \in I} F_i$? +Back: A function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the domain of $F$ assumed to be in expression $\bigcup_{i \in I} F(i)$? +Back: Some superset of $I$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What condition must $I$ satisfy in expression $\bigcup_{i \in I} F(i)$? +Back: N/A. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Suppose $I = \{0, 1, 2\}$. What does $\bigcup_{i \in I} F(i)$ evaluate to? +Back: $F(0) \cup F(1) \cup F(2)$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Suppose $I = \varnothing$. What does $\bigcup_{i \in I} F(i)$ evaluate to? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What kind of mathematic object is $F$ in expression $\bigcap_{i \in I} F(i)$? +Back: A function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is $\bigcap_{i \in I} F(i)$ often alternatively denoted? +Back: $\bigcap_{i \in I} F_i$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the domain of $F$ assumed to be in expression $\bigcap_{i \in I} F(i)$? +Back: Some superset of $I$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What condition must $I$ satisfy in expression $\bigcap_{i \in I} F(i)$? +Back: $I \neq \varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Suppose $I = \{0, 1, 2\}$. What does $\bigcap_{i \in I} F(i)$ evaluate to? +Back: $F(0) \cap F(1) \cap F(2)$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Suppose $I = \varnothing$. What does $\bigcap_{i \in I} F(i)$ evaluate to? +Back: N/A. This is undefined. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is $\bigcap_{i \in I} F_i$ alternatively denoted? +Back: $\bigcap_{i \in I} F(i)$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What kind of mathematic object is $F$ in expression $\bigcap_{i \in I} F_i$? +Back: A function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +## Function Sets + +For sets $A$ and $B$, the collection of functions $F$ from $A$ into $B$ is: $$^AB = \{F \mid F \colon A \rightarrow B\}$$ +$^AB$ is read as "$B$-pre-$A$". It is often written as $B^A$ instead. + +%%ANKI +Basic +For sets $A$ and $B$, how is set $B^A$ defined? +Back: $\{F \mid F \colon A \rightarrow B\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For sets $A$ and $B$, how is set $^AB$ defined? +Back: $\{F \mid F \colon A \rightarrow B\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For any function $F \colon A \rightarrow B$, $F$ is a subset of what other set? +Back: $A \times B$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For any function $F \colon A \rightarrow B$, $F$ is a member of what other set? +Back: $\mathscr{P}(A \times B)$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For sets $A$ and $B$, how is set $B^A$ pronounced? +Back: As "$B$-pre-$A$". +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Why prefer notation $B^A$ over $^AB$? +Back: The notation mirrors $|B|^{|A|}$, the number of elements in $B^A$ given both sets are finite. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For sets $A$ and $B$, how is set $^AB$ pronounced? +Back: As "$B$-pre-$A$". +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Why prefer notation $^AB$ over $B^A$? +Back: Because the sets are written left-to-right, from domain to codomain. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +*Why* is set $B^A$ denoted the way it is? +Back: If $A$ and $B$ are finite, then $B^A$ has $|B|^{|A|}$ members. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the domain of $^\omega\{0, 1\}$? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the domain of a member of $^\omega\{0, 1\}$? +Back: $\omega$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the range of $\{0, 1\}^\omega$? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the range of a member of $\{0, 1\}^\omega$? +Back: $\{0, 1\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What does $\varnothing^\varnothing$ evaluate to? +Back: $\{\varnothing\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For $A \neq \varnothing$, what does $\varnothing^A$ evaluate to? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For $A \neq \varnothing$, *why* does $\varnothing^A = \varnothing$? +Back: No function can map a nonempty domain to an empty range. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For $A \neq \varnothing$, what does $^\varnothing A$ evaluate to? +Back: $\{\varnothing\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +For $A \neq \varnothing$, *why* does $^\varnothing A = \{\varnothing\}$? +Back: $\varnothing$ is the only function with empty domain. +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). \ No newline at end of file diff --git a/notes/hashing/index.md b/notes/hashing/index.md index dafbc61..8178ab2 100644 --- a/notes/hashing/index.md +++ b/notes/hashing/index.md @@ -108,7 +108,7 @@ END%% %%ANKI Basic What does a hash table collision refer to? -Back: Two keys hashing to the same slot. +Back: Two different keys hashing to the same slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% diff --git a/notes/lambda-calculus/beta-reduction.md b/notes/lambda-calculus/beta-reduction.md index c49f773..38e9a5f 100644 --- a/notes/lambda-calculus/beta-reduction.md +++ b/notes/lambda-calculus/beta-reduction.md @@ -559,7 +559,7 @@ END%% %%ANKI Basic -If a $\lambda$-term has $\beta$-normal forms $P$ and $Q$, what can be said about $P$ and $Q$? +If a $\lambda$-term has $\beta$-normal forms $P$ and $Q$, how do $P$ and $Q$ relate to one another? Back: $P \equiv_\alpha Q$ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).