Hashing, relations, VCs.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

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-    "programming/pred-trans.md": "611dfb04df2126d35d67ca7ca1b39f52",
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-    "hashing/direct-addressing.md": "acff356a70980cb4135094fccb3b3187",
+    "hashing/direct-addressing.md": "f75cc22e74ae974fe4f568a2ee9f951f",
-    "hashing/index.md": "2f75e951aac609a2a9da731ef5444214",
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-    "lambda-calculus/alpha-conversion.md": "d3a10cb834e696909b04611c83f738cb",
+    "lambda-calculus/alpha-conversion.md": "e87b499517c2471cae4717703ca3aba0",
     "lambda-calculus/index.md": "756c93b8717fd00b04f8a99509066486",
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     "x86-64/instructions/arithmetic.md": "271218d855e7291f119f96e91f582738",
-    "x86-64/instructions/access.md": "ab8605e11c0a3936735b18993e215513",
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-    "hashing/closed-addressing.md": "962a48517969bf5e410cf78fc584051f",
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-    "lambda-calculus/beta-reduction.md": "aa1b302755cde85085abedbde85161df"
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   },
   "fields_dict": {
     "Basic": [

notes/_journal/2024-06-16.md

@@ -1,9 +0,0 @@
----
-title: "2024-06-16"
----
-
-- [ ] Anki Flashcards
-- [ ] KoL
-- [ ] OGS
-- [ ] Sheet Music (10 min.)
-- [ ] Korean (Read 1 Story)

notes/_journal/2024-06-21.md

@@ -0,0 +1,12 @@
+---
+title: "2024-06-21"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[functions#Inverses|inverses]] and [[functions#Compositions|compositions]] of functions.
+* Finished Chapter 8 of "The Science of Programming" on sequential composition.

notes/_journal/2024-06/2024-06-16.md

@@ -0,0 +1,9 @@
+---
+title: "2024-06-16"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2024-06/2024-06-17.md

@@ -0,0 +1,9 @@
+---
+title: "2024-06-17"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2024-06/2024-06-18.md

@@ -0,0 +1,12 @@
+---
+title: "2024-06-18"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Preliminary read on condition codes in "Computer Systems: A Programmer's Perspective".
+* Notes on the load factor of open/closed addressing hash tables.

notes/_journal/2024-06/2024-06-19.md

@@ -0,0 +1,11 @@
+---
+title: "2024-06-19"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Read through the introduction of "Venture Deals"

notes/_journal/2024-06/2024-06-20.md

@@ -0,0 +1,11 @@
+---
+title: "2024-06-20"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Read Chapter 1 of "Venture Deals". Begin note-taking on [[venture-capitalist|VCs]].

notes/c17/strings.md

@@ -651,7 +651,7 @@ END%%
 
 %%ANKI
 Basic
-How many digits follows `e` in the output of `printf` specifier `%e`?
+How many digits follow `e` in the output of `printf` specifier `%e`?
 Back: At least `2`.
 Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
 Tags: printf

notes/hashing/closed-addressing.md

@@ -49,6 +49,104 @@ Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” acc
 <!--ID: 1718198755496-->
 END%%
 
+%%ANKI
+Basic
+What is the theoretical maximum load factor in closed addressing?
+Back: N/A
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718759188231-->
+END%%
+
+%%ANKI
+Basic
+*Why* is the theoretical maximum load factor of closed addressing unbounded?
+Back: A closed addressing hash table can always have more entries inserted into it.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718759188234-->
+END%%
+
+%%ANKI
+Basic
+When is the load factor of a closed addressing hash table $0$?
+Back: When no entries are stored in the table.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188238-->
+END%%
+
+%%ANKI
+Basic
+When is the load factor of a closed addressing hash table $1$?
+Back: When there exist the same number of total entries as slots.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188241-->
+END%%
+
+%%ANKI
+Basic
+When is the load factor of a closed addressing hash table $> 1$?
+Back: When there exist more total entries than number of slots.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188245-->
+END%%
+
+## Chaining
+
+The most common form of closed addressing is **chaining**. In this scheme, each slot $j$ is a (nullable) pointer to the head of a linked list containing all the elements with hash value $j$.
+
+%%ANKI
+Basic
+What is the most common implementation of closed addressing?
+Back: Chaining.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188249-->
+END%%
+
+%%ANKI
+Basic
+What data structure is typically used in a hash table with chaining?
+Back: Linked lists.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188252-->
+END%%
+
+%%ANKI
+Basic
+Consider a hash table with chaining. What is in an empty slot?
+Back: A NIL pointer.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188256-->
+END%%
+
+%%ANKI
+Basic
+Consider a hash table with chaining. What is in a nonempty slot?
+Back: A pointer to the head of a linked list.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188261-->
+END%%
+
+%%ANKI
+Basic
+Consider a hash table with chaining. How many linked list instances exist?
+Back: One for each slot in the hash table.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188269-->
+END%%
+
+%%ANKI
+Cloze
+A hash table with chaining is an example of {closed} addressing.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188275-->
+END%%
+
+%%ANKI
+Cloze
+A hash table with chaining is an example of {open} hashing.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188281-->
+END%%
+
 ## Bibliography
 
 * “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).

notes/hashing/direct-addressing.md

@@ -137,6 +137,14 @@ Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” acc
 <!--ID: 1718199205872-->
 END%%
 
+%%ANKI
+Basic
+What is the theoretical maximum load factor in direct addressing?
+Back: $1$
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718759188227-->
+END%%
+
 ## Bibliography
 
 * “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).

notes/hashing/index.md

@@ -137,6 +137,68 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1716307180980-->
 END%%
 
+Consider hash table $T$ with $m$ slots that stores $n$ entries. Then the **load factor** $\alpha$ for $T$ is defined to be $n / m$, i.e. the average number of entries that map to the same slot.
+
+%%ANKI
+Basic
+The load factor is a ratio of what two numbers?
+Back: The number of entries in the table to the number of slots stored in the table.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188190-->
+END%%
+
+%%ANKI
+Cloze
+The load factor of a hash table {increases} as the number of slots {decrease}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188194-->
+END%%
+
+%%ANKI
+Cloze
+The load factor of a hash table {decreases} as the number of total entries {decrease}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188199-->
+END%%
+
+%%ANKI
+Cloze
+The load factor of a hash table {increases} as the number of total entries {increase}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188204-->
+END%%
+
+%%ANKI
+Cloze
+The load factor of a hash table {decreases} as the number of slots {increase}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188208-->
+END%%
+
+%%ANKI
+Basic
+Let $n / m$ denote the load factor of a hash table. What does $n$ represent?
+Back: The total number of entries in the table.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188214-->
+END%%
+
+%%ANKI
+Basic
+Let $n / m$ denote the load factor of a hash table. What does $m$ represent?
+Back: The number of slots in the table.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188218-->
+END%%
+
+%%ANKI
+Basic
+*Why* is the load factor $\alpha$ of a hash table defined the way it is?
+Back: It represents the average number of entries stored at a slot.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188222-->
+END%%
+
 An **independent uniform hash function** is the ideal theoretical abstraction. For each possible input $k$ in universe $U$, an output $h(k)$ is produced randomly and independently chosen from range $\{0, 1, \ldots, m - 1\}$. Once a value $h(k)$ is chosen, each subsequent call to $h$ with the same input $k$ yields the same output $h(k)$.
 
 %%ANKI

notes/hashing/open-addressing.md

@@ -49,6 +49,46 @@ Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” acc
 <!--ID: 1718198755486-->
 END%%
 
+%%ANKI
+Basic
+What is the theoretical maximum load factor in open addressing?
+Back: $1$
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718759188171-->
+END%%
+
+%%ANKI
+Basic
+*Why* is the theoretical maximum load factor of open addressing unbounded?
+Back: An open addressing hash table can only store as many entries as slots.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718759188176-->
+END%%
+
+%%ANKI
+Basic
+When is the load factor of an open addressing hash table $0$?
+Back: When no entries are stored in the table.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188179-->
+END%%
+
+%%ANKI
+Basic
+When is the load factor of a open addressing hash table $1$?
+Back: When there exist the same number of total entries as slots.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188182-->
+END%%
+
+%%ANKI
+Basic
+When is the load factor of an open addressing hash table $> 1$?
+Back: N/A
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718759188186-->
+END%%
+
 ## Bibliography
 
 * “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).

notes/lambda-calculus/alpha-conversion.md

@@ -62,10 +62,10 @@ END%%
 
 %%ANKI
 Basic
-What property must $y$ satisfy for $\lambda x. M \equiv_\alpha \lambda y. M$?
+What property must $y$ satisfy for $\lambda x. M \equiv_\alpha \lambda y. [y/x]M$?
 Back: $y \not\in FV(M)$
 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).
-<!--ID: 1717687744147-->
+<!--ID: 1718802166425-->
 END%%
 
 %%ANKI
@@ -336,7 +336,7 @@ END%%
 
 %%ANKI
 Cloze
-{$F$} $\Rightarrow [P/x][Q/x]M \equiv_\alpha [([P/x]Q)/x]M$
+{$T$} $\Rightarrow [P/x][Q/x]M \equiv_\alpha [([P/x]Q)/x]M$
 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).
 <!--ID: 1718422235912-->
 END%%
@@ -402,7 +402,7 @@ For $\lambda$-terms $M$, $M'$, $N$, and $N'$, and variable $x$, $$M \equiv_\alph
 %%ANKI
 Basic
 The proof of which implication shows "substitution is well-behaved w.r.t. $\alpha$-conversion"?
-Back: $M \equiv_\alpha M' \land N \equiv_\alpha N' \Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$
+Back: $P \equiv_\alpha P' \land M \equiv_\alpha M' \Rightarrow [P/x]M \equiv_\alpha [P'/x]M'$
 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).
 <!--ID: 1718422973129-->
 END%%

notes/logic/prop-logic.md

@@ -335,7 +335,7 @@ END%%
 
 %%ANKI
 Basic
-Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
+Is $(b \land c)$ well-defined in $\{\langle b, T \rangle, \langle c, F \rangle\}$?
 Back: Yes.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1706994861318-->
@@ -343,7 +343,7 @@ END%%
 
 %%ANKI
 Basic
-Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
+Is $(b \lor d)$ well-defined in $\{\langle b, T \rangle, \langle c, F \rangle\}$?
 Back: No.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1706994861320-->
@@ -368,7 +368,7 @@ END%%
 %%ANKI
 Basic
 What set of states does proposition $a \land b$ represent?
-Back: $\{\{(a, T), (b, T)\}\}$
+Back: $\{\{\langle a, T \rangle, \langle b, T \rangle\}\}$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1706994861339-->
 END%%
@@ -376,7 +376,7 @@ END%%
 %%ANKI
 Basic
 What set of states does proposition $a \lor b$ represent?
-Back: $\{\{(a, T), (b, T)\}, \{(a, T), (b, F)\}, \{(a, F), (b, T)\}\}$
+Back: $\{\{\langle a, T \rangle, \langle b, T \rangle\}, \{\langle a, T \rangle, \langle b, F \rangle\}, \{\langle a, F \rangle, \langle b, T \rangle\}\}$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1715895996324-->
 END%%

notes/logic/quantification.md

@@ -190,29 +190,6 @@ END%%
 
 Identifiers are said to be **bound** if they are parameters to a quantifier. Identifiers that are not bound are said to be **free**. A first-order logic formula is said to be in **prenex normal form** (PNF) if written in two parts: the first consisting of quantifiers and bound variables (the **prefix**), and the second consisting of no quantifiers (the **matrix**).
 
-%%ANKI
-Basic
-When is an identifier said to be bound?
-Back: When it is specified as a parameter to a quantifier.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1707674796768-->
-END%%
-
-%%ANKI
-Basic
-When is an identifier said to be free?
-Back: When it isn't specified as a parameter to a quantifier.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1707674796770-->
-END%%
-
-%%ANKI
-Cloze
-An identifier that is not {bound} is instead {free}.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1707674796772-->
-END%%
-
 %%ANKI
 Basic
 Prenex normal form consists of what two parts?

notes/programming/pred-trans.md

@@ -55,7 +55,7 @@ END%%
 
 %%ANKI
 Basic
-What is the antecedent of $\{Q\}\; S\; \{R\}$ in English?
+Interpret $\{Q\}\; S\; \{R\}$ in English. What is the antecedent of the implication?
 Back: $S$ is executed in a state satisfying $Q$.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1714420640229-->
@@ -63,7 +63,7 @@ END%%
 
 %%ANKI
 Basic
-What is the consequent of $\{Q\}\; S\; \{R\}$ in English?
+Interpret $\{Q\}\; S\; \{R\}$ in English. What is the consequent of the implication?
 Back: $S$ terminates in a finite amount of time in a state satisfying $R$.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1714420640231-->
@@ -504,8 +504,8 @@ For any predicate $R$, $wp(skip, R) = R$.
 
 %%ANKI
 Basic
-How is the $skip$ command defined?
+How is the $skip$ command defined in terms of $wp$?
-Back: As $wp(skip, R) = R$.
+Back: For any predicate $R$, $wp(skip, R) = R$.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1716810300099-->
 END%%
@@ -531,8 +531,8 @@ For any predicate $R$, $wp(abort, R) = F$.
 
 %%ANKI
 Basic
-How is the $abort$ command defined?
+How is the $abort$ command defined in terms of $wp$?
-Back: As $wp(abort, R) = F$.
+Back: For any predicate $R$, $wp(abort, R) = F$.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1716810300116-->
 END%%
@@ -584,6 +584,42 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1716810300145-->
 END%%
 
+### Sequential Composition
+
+**Sequential composition** is one way of composing larger program segments from smaller segments. Let $S1$ and $S2$ be two commands. Then $S1; S2$ is defined as $$wp(''S1; S2'', R) = wp(S1, wp(S2, R))$$
+
+%%ANKI
+Basic
+Let $S1$ and $S2$ be two commands. How is their sequential composition denoted?
+Back: $S1; S2$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1719019485648-->
+END%%
+
+%%ANKI
+Basic
+How is $S1; S2$ defined in terms of $wp$?
+Back: For any predicate $R$, $wp(''S1; S2'', R) = wp(S1, wp(S2, R))$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1719019485654-->
+END%%
+
+%%ANKI
+Basic
+Is sequential composition commutative?
+Back: No.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1719019485662-->
+END%%
+
+%%ANKI
+Basic
+Is sequential composition associative?
+Back: Yes.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1719019485666-->
+END%%
+
 ## Bibliography
 
 * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

notes/set/functions.md

@@ -328,7 +328,7 @@ END%%
 
 %%ANKI
 Cloze
-Let $F$ be a function. Then $F$ maps {$\mathop{\text{dom}}F$} onto {$\mathop{\text{ran}}F$}.
+Let $F$ be a function. Then $F$ maps {$\mathop{\text{dom} }F$} onto {$\mathop{\text{ran} }F$}.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1718427443419-->
 END%%
@@ -364,7 +364,7 @@ END%%
 Basic
 *Why* isn't the following a surjection?
 ![[function-general.png]]
-Back: No element of $X$ maps to $b \in Y$.
+Back: No element of $X$ maps to $a$ or $b$.
 Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
 <!--ID: 1718465870573-->
 END%%
@@ -438,6 +438,244 @@ Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 202
 <!--ID: 1718465870605-->
 END%%
 
+## Inverses
+
+Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}.$$
+%%ANKI
+Basic
+What kind of mathematical object does the inverse operation apply to?
+Back: Sets.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770704-->
+END%%
+
+%%ANKI
+Basic
+What is the "arity" of the inverse operation in set theory?
+Back: $1$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017251246-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ be a set. How is the inverse of $F$ denoted?
+Back: $F^{-1}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770741-->
+END%%
+
+%%ANKI
+Basic
+What kind of mathematical object does the inverse operation emit?
+Back: Relations.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770749-->
+END%%
+
+%%ANKI
+Basic
+How is the inverse of set $F$ defined in set-builder notation?
+Back: $F^{-1} = \{\langle u, v \rangle \mid vFu\}$\
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770752-->
+END%%
+
+%%ANKI
+Basic
+Consider set $A$. Is $A^{-1}$ a relation?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770755-->
+END%%
+
+%%ANKI
+Basic
+Consider set $A$. Is $A^{-1}$ a function?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770759-->
+END%%
+
+%%ANKI
+Basic
+Consider relation $R$. Is $R^{-1}$ a relation?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770763-->
+END%%
+
+%%ANKI
+Basic
+Consider relation $R$. Is $R^{-1}$ a function?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770767-->
+END%%
+
+%%ANKI
+Basic
+Consider function $F \colon A \rightarrow B$. Is $F^{-1}$ a relation?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770772-->
+END%%
+
+%%ANKI
+Basic
+Consider function $F \colon A \rightarrow B$. Is $F^{-1}$ a function?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770778-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ a function?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770782-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ one-to-one?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770787-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$ be an injection. Is $F^{-1}$ onto $A$?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770792-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$ be a surjection. Is $F^{-1}$ a function?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770796-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$ be a surjection. Is $F^{-1}$ a relation?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770800-->
+END%%
+
+%%ANKI
+Basic
+Consider function $F \colon A \rightarrow B$. What is the domain of $F^{-1}$?
+Back: $\mathop{\text{ran}}F$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770805-->
+END%%
+
+%%ANKI
+Basic
+Consider function $F \colon A \rightarrow B$. What is the range of $F^{-1}$?
+Back: $A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016770812-->
+END%%
+
+%%ANKI
+Basic
+Consider function $F$. How does $(F^{-1})^{-1}$ relate to $F$?
+Back: They are equal.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016946539-->
+END%%
+
+%%ANKI
+Basic
+Consider relation $R$. How does $(R^{-1})^{-1}$ relate to $R$?
+Back: They are equal.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016946547-->
+END%%
+
+%%ANKI
+Basic
+Consider set $A$. How does $(A^{-1})^{-1}$ relate to $A$?
+Back: $(A^{-1})^{-1}$ is a subset of $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719016946554-->
+END%%
+
+%%ANKI
+Basic
+When does $A \neq (A^{-1})^{-1}$?
+Back: If there exists an $x \in A$ such that $x$ is not an ordered pair.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017560113-->
+END%%
+
+## Compositions
+
+Let $F$ and $G$ be arbitrary sets. The **composition** of $F$ and $G$ is the set $$F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv \}$$
+
+%%ANKI
+Basic
+What kind of mathematical object does the composition operation apply to?
+Back: Sets.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017251256-->
+END%%
+
+%%ANKI
+Basic
+What kind of mathematical object does the composition operation emit?
+Back: Relations.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017251259-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ and $G$ be arbitrary sets. How is the composition of $G$ and $F$ denoted?
+Back: $G \circ F$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017251252-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ and $G$ be arbitrary sets. How is the composition of $F$ and $G$ denoted?
+Back: $F \circ G$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017251262-->
+END%%
+
+%%ANKI
+Basic
+What is the "arity" of the composition operation in set theory?
+Back: $2$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017251265-->
+END%%
+
+%%ANKI
+Cloze
+{$(F \circ G)(x)$} is alternatively written as {$F(G(x))$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017560120-->
+END%%
+
+%%ANKI
+Basic
+How is the composition of sets $F$ and $G$ defined in set-builder notation?
+Back: $F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719017560123-->
+END%%
+
 ## Bibliography
 
 * “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).

notes/set/graphs.md

@@ -1061,7 +1061,7 @@ END%%
 %%ANKI
 Basic
 What is the codomain of an isomorphism between graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$?
-Back: $V_2$.
+Back: $V_2$
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 <!--ID: 1715537560183-->
 END%%

notes/set/relations.md

@@ -189,6 +189,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1718107987862-->
 END%%
 
+%%ANKI
+Basic
+What is the most general mathematical object the $\mathop{\text{dom}}$ operation can be applied to?
+Back: Sets.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718546439334-->
+END%%
+
 %%ANKI
 Basic
 Let $A$ be a set containing no ordered pairs. What is $\mathop{\text{dom}} A$?
@@ -281,6 +289,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1718107987880-->
 END%%
 
+%%ANKI
+Basic
+What is the most general mathematical object the $\mathop{\text{ran}}$ operation can be applied to?
+Back: Sets.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718546439338-->
+END%%
+
 %%ANKI
 Basic
 Let $A$ be a set containing no ordered pairs. What is $\mathop{\text{ran}} A$?
@@ -337,6 +353,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1718327739955-->
 END%%
 
+%%ANKI
+Basic
+What is the most general mathematical object the $\mathop{\text{fld}}$ operation can be applied to?
+Back: Sets.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718546439341-->
+END%%
+
 %%ANKI
 Basic
 $\mathop{\text{fld}} R = \bigcup \bigcup R$ is necessary for what condition?

notes/startups/index.md

@@ -0,0 +1,7 @@
+---
+title: Startups
+TARGET DECK: Obsidian::H&SS
+FILE TAGS: startups
+tags:
+  - startups
+---

notes/startups/venture-capitalist.md

@@ -0,0 +1,190 @@
+---
+title: Venture Capitalist
+TARGET DECK: Obsidian::H&SS
+FILE TAGS: startups::vc
+tags:
+  - startups
+---
+
+## Overview
+
+A **venture capitalist** (VC) is an investor who provides capital to companies in exchange for an equity stake on behalf of a firm. A firm comprises of the following roles (in order of seniority):
+
+* **Managing director** (MD) or **general partner** (GP). The VCs that make the final investment decisions and sit on the boards of directors of the companies they invest in.
+* **Principal** or **director**. Junior deal professionals looking to become managing directors.
+* **Associate**. Work for one or more deal partners, usually a managing director.
+* **Analyst**. Individuals with similar responsibilites as the associate, though usually less deal-centric.
+
+%%ANKI
+Basic
+What is VC short for?
+Back: **V**enture **c**apitalist.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788273-->
+END%%
+
+%%ANKI
+Basic
+What ambiguity does the term "VC" introduce?
+Back: It may refer to a VC firm or an individual of said firm.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788303-->
+END%%
+
+%%ANKI
+Cloze
+Typically VCs provide {capital} in exchange for {equity}.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788311-->
+END%%
+
+%%ANKI
+Basic
+How is a "venture capitalist" defined?
+Back: An investor who provides capital to companies, on behalf of a firm, in exchange for equity.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788328-->
+END%%
+
+%%ANKI
+Basic
+What form of capital does a VC typically work in?
+Back: Cash flow.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788334-->
+END%%
+
+%%ANKI
+Basic
+Within a VC firm, what does MD stand for?
+Back: **M**anaging **d**irector.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788338-->
+END%%
+
+%%ANKI
+Basic
+Within a VC firm, what does GP stand for?
+Back: **G**eneral **p**artner.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788342-->
+END%%
+
+%%ANKI
+Basic
+With respect to a VC firm, what does a "managing director" refer to?
+Back: A senior VC, generally responsible for making final investment decisions.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311552-->
+END%%
+
+%%ANKI
+Basic
+With respect to a VC firm, what does a "general partner" refer to?
+Back: A senior VC, generally responsible for making final investment decisions.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788345-->
+END%%
+
+%%ANKI
+Cloze
+A {general partner} is also known as a {managing director}.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788348-->
+END%%
+
+%%ANKI
+Cloze
+The {principal/director} role follows the {MD/GP} role in seniority.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788353-->
+END%%
+
+%%ANKI
+Basic
+With respect to a VC firm, what does a "principal" refer to?
+Back: A VC working their way up to becoming a GP.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788357-->
+END%%
+
+%%ANKI
+Basic
+With respect to a VC firm, what does a "director" refer to?
+Back: A VC working their way up to becoming an MD.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311558-->
+END%%
+
+%%ANKI
+Cloze
+A {principal} is also known as a {director}.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788361-->
+END%%
+
+%%ANKI
+Basic
+What types of VCs are grouped under term "deal partner"?
+Back: GPs and principals.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718878788368-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes VCs from angel investors?
+Back: The former use a pool of investors' money. The latter uses their own money.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311563-->
+END%%
+
+%%ANKI
+Cloze
+The {associate} role follows the {principal/director} role in seniority.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311566-->
+END%%
+
+%%ANKI
+Cloze
+The {analyst} role follows the {associate} role in seniority.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311572-->
+END%%
+
+%%ANKI
+Basic
+How are analysts and associates typically distinguished?
+Back: The latter are usually more deal-centric than the former.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311577-->
+END%%
+
+%%ANKI
+Basic
+What role is a recent college graduate likely given at a VC firm?
+Back: Analyst.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311581-->
+END%%
+
+%%ANKI
+Basic
+With respect to a VC firm, what does an "associate" refer to?
+Back: An employee usually working directly for one or more deal managers.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311586-->
+END%%
+
+%%ANKI
+Basic
+With respect to a VC firm, what does an "analyst" refer to?
+Back: An employee working on general functions for the firm.
+Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
+<!--ID: 1718879311590-->
+END%%
+
+## Bibliography
+
+* Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.

notes/x86-64/instructions/access.md

@@ -348,7 +348,7 @@ Basic
 How is `pushq %rbp` equivalently written using a pair of instructions?
 Back:
 ```asm
-subq 8,%rsp
+subq $8,%rsp
 movq %rbp,(%rsp)
 ```
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
@@ -361,7 +361,7 @@ How is `popq %rax` equivalently written using a pair of instructions?
 Back:
 ```asm
 movq (%rsp),%rax
-addq 8,%rsp
+addq $8,%rsp
 ```
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1715377284962-->