Topological sort and NFAs.

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

@@ -233,7 +233,9 @@
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notes/_journal/2025-01-17.md

@@ -0,0 +1,9 @@
+---
+title: "2025-01-17"
+---
+
+- [ ] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2025-01/2025-01-13.md

@@ -0,0 +1,11 @@
+---
+title: "2025-01-13"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Formal definition of [[automaton#Nondeterminism|NFA]]s.

notes/_journal/2025-01/2025-01-14.md

@@ -0,0 +1,11 @@
+---
+title: "2025-01-14"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* [[integrals#Integrable Functions|Integrals]] of more general functions.

notes/_journal/2025-01/2025-01-15.md

@@ -0,0 +1,9 @@
+---
+title: "2025-01-15"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2025-01/2025-01-16.md

@@ -0,0 +1,11 @@
+---
+title: "2025-01-16"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[dfs#Topological Sort|topological sorting]].

notes/algorithms/dfs.md

@@ -614,6 +614,128 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1735785623439-->
 END%%
 
+## Topological Sort
+
+A topological sort of a directed acyclic graph $G$ is an ordering of all its vertices such that if $G$ contains an edge $\langle u, v \rangle$, then $u$ appears before $v$ in the ordering.
+
+> Call depth-first search on $G$ to compute finish times $v{.}f$ for each vertex $v$. As each vertex is finished, insert it onto the front of a linked list. Return the list when all vertices are processed.
+
+%%ANKI
+Basic
+*What* is a topological sort?
+Back: An ordering of vertices such that if a DAG has edge $\langle u, v \rangle$, then $u$ appears before $v$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266647-->
+END%%
+
+%%ANKI
+Basic
+Which basic graph algorithm is used in toplogical sorting?
+Back: Depth-first search.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266658-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't toplogical sort applicable to digraphs with cycles?
+Back: In the case of cycles, there is no notion of a vertex coming before another.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266662-->
+END%%
+
+%%ANKI
+Basic
+Let $G$ be a DAG with edge $\langle u, v \rangle$. How do $u$ and $v$ relate in $G$'s topological sort?
+Back: $u$ comes before $v$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266665-->
+END%%
+
+%%ANKI
+Basic
+Let $G$ be a DAG with path $\langle v_1, \ldots, v_n \rangle$. How do $v_1$ and $v_n$ relate in $G$'s topological sort?
+Back: $v_1$ comes before $v_n$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266668-->
+END%%
+
+%%ANKI
+Basic
+What kind of graph is a topological sort applicable to?
+Back: A directed acyclic graph.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266673-->
+END%%
+
+%%ANKI
+Basic
+How many topological sorts might a DAG have?
+Back: One or more.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266676-->
+END%%
+
+%%ANKI
+Basic
+DFS on a DAG cannot produce what edge classification?
+Back: Back edges.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266680-->
+END%%
+
+%%ANKI
+Cloze
+A directed graph is {acyclic} if and only if DFS produces no {back} edges.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266684-->
+END%%
+
+%%ANKI
+Basic
+Describe how the toplogical sort algorithm on a directed acyclic graph $G$ is performed.
+Back: Run DFS. As each vertex is finished processing, prepend it to a return list.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266688-->
+END%%
+
+%%ANKI
+Basic
+Assuming an adjacency-list represention of a DAG, what is topological sort's runtime?
+Back: $\Theta(\lvert V \rvert + \lvert E \rvert)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266692-->
+END%%
+
+%%ANKI
+Basic
+Assuming an adjacency-matrix represention of a DAG, what is topological sort's runtime?
+Back: $\Theta(\lvert V \rvert^2)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266695-->
+END%%
+
+%%ANKI
+Basic
+How many topological sorts exist in the following graph?
+![[topological-sort.png]]
+Back: Three.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266699-->
+END%%
+
+%%ANKI
+Basic
+What are the possible topological sorts of the following graph?
+![[topological-sort.png]]
+Back:
+1. `B -> D -> A -> C`
+2. `A -> B -> D -> C`
+3. `A -> B -> C -> D`
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1737086266703-->
+END%%
+
 ## Bibliography
 
 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/calculus/integrals.md

@@ -238,7 +238,192 @@ END%%
 
 ## Integrable Functions
 
-TODO
+Let $f$ be a function defined and bounded on $[a, b]$. Let $s$ and $t$ denote arbitrary step functions defined on $[a, b]$ such that $s(x) \leq f(x) \leq t(x)$ for all $x \in [a, b]$. If for every such $s$ and $t$, there is exactly one number $I$ satisfying $$\int_a^b s(x) \,dx \leq I \leq \int_a^b t(x) \,dx,$$
+then $I$ is said to be the **integral of $f$ from $a$ to $b$** and is denoted by symbol $\int_a^b f(x) \,dx$. When such an $I$ exists, the function $f$ is said to be **integrable** on $[a, b]$.
+
+%%ANKI
+Basic
+The integral of a function is approximated above and below by integrals of what kind of functions?
+Back: Step functions.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432232-->
+END%%
+
+%%ANKI
+Basic
+How is the integral of $f$ from $a$ to $b$ denoted?
+Back: As $\int_a^b f(x) \,dx$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+END%%
+
+%%ANKI
+Basic
+*Why* does Apostol only consider integrals of functions that are bounded over an interval?
+Back: Because the integral is defined by approximating step functions above and below the function.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432249-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for function $f$ to be bounded on $[a, b]$?
+Back: There exists some $M > 0$ such that $-M \leq f(x) \leq M$ for all $x \in [a, b]$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432252-->
+END%%
+
+%%ANKI
+Cloze
+The {integral of $f$ from $a$ to $b$} is denoted as {$\int_a^b f(x) \,dx$}.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432255-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for step function $s$ to be below function $f$ on $[a, b]$?
+Back: That $s(x) \leq f(x)$ for all $x \in [a, b]$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432259-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for step function $s$ to be above function $f$ on $[a, b]$?
+Back: That $f(x) \leq s(x)$ for all $x \in [a, b]$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432262-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function. Let $s$ and $t$ be step functions such that $s$ is above $f$ and $t$ is below $f$. What inequality arises?
+Back: $t \leq f \leq s$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432266-->
+END%%
+
+%%ANKI
+Cloze
+Let $f$ be a function defined and {bounded} on $[a, b]$. Let $s$ and $t$ denote {arbitrary step functions} such that {$$s(x) \leq f(x) \leq t(x)$$} for all $x \in [a, b]$. If for every such $s$ and $t$, there is {exactly one} $I$ such that {$$\int_a^b s(x) \,dx \leq I \leq \int_a^b t(x) \,dx,$$} then $I$ is said to be the {integral of $f$ from $a$ to $b$}.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432271-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. What must there exist exactly one of for $f$ to be integrable on $[a, b]$?
+Back: A number $I$ such that $\int_a^b s(x) \,dx \leq I \leq \int_a^b t(x) \,dx$ for all step functions $s$ and $t$ satisfying $s \leq f \leq t$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432277-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. What does it mean for $f$ to be integrable on $[a, b]$?
+Back: There exists exactly one number $I$ such that $\int_a^b s(x) \,dx \leq I \leq \int_a^b t(x) \,dx$ for all step functions $s$ and $t$ satisfying $s \leq f \leq t$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736892432282-->
+END%%
+
+The **lower integral** of $f$, denoted by $\underline{I}(f)$, is defined as $$\underline{I}(f) = \mathop{\text{sup}} \left\{ \int_a^b s(x) \,dx \mid s \leq f \right\}.$$
+Likewise, the **upper integral** of $f$, denoted by $\bar{I}(f)$, is defined as $$\bar{I}(f) = \mathop{\text{inf}} \left\{ \int_a^b t(x) \,dx \mid f \leq t \right\}.$$
+Thus $f$ is integrable on $[a, b]$ if and only if $\int_a^b f(x) \,dx = \underline{I}(f) = \bar{I}(f)$.
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. How is the lower integral of $f$ denoted?
+Back: $\underline{I}(f)$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396749-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. What does $\underline{I}(f)$ denote?
+Back: The lower integral of $f$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893441504-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. How is the upper integral of $f$ denoted?
+Back: $\bar{I}(f)$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396757-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. What does $\bar{I}(f)$ denote?
+Back: The upper integral of $f$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893441509-->
+END%%
+
+%%ANKI
+Cloze
+Let $f$ be a function defined and bounded on $[a, b]$. Then {1: $\underline{I}(f)$} is to a {2:supremum} whereas {2:$\bar{I}(f)$} is to an {1:infimum}.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396764-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. How do we know the following set is nonempty? $$\left\{ \int_a^b t(x) \, dx \mid s \text{ is a step function below } f \right\}$$
+Back: It's nonempty because $f$ is bounded.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396770-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. $\underline{I}(f)$ equals the supremum of what set?
+Back: $\left\{ \int_a^b s(x) \, dx \mid s \text{ is a step function below } f \right\}$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396776-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. $\underline{I}(f)$ equals the infimum of what set?
+Back: N/A. The lower integral is a supremum.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396782-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. How do we know the following set is nonempty? $$\left\{ \int_a^b t(x) \, dx \mid t \text{ is a step function above } f \right\}$$
+Back: It's nonempty because $f$ is bounded.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396788-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. $\bar{I}(f)$ equals the supremum of what set?
+Back: N/A. The upper integral is an infimum.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396794-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. $\bar{I}(f)$ equals the infimum of what set?
+Back: $\left\{ \int_a^b t(x) \, dx \mid t \text{ is a step function above } f \right\}$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396799-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be a function defined and bounded on $[a, b]$. If $s$ and $t$ are step functions s.t. $s \leq f \leq t$, what integral property guarantees $\int_a^b s(x) \,dx \leq \int_a^b t(x) \,dx$?
+Back: The comparison theorem.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1736893396804-->
+END%%
 
 ### Integrand Additivity
 

notes/computability/automaton.md

@@ -125,6 +125,20 @@ Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third ed
 <!--ID: 1735160593029-->
 END%%
 
+%%ANKI
+Cloze
+{1:Edges} are to {2:graphs} whereas {2:transitions} are to {1:state diagrams}.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1734999643325-->
+END%%
+
+%%ANKI
+Cloze
+{1:Vertices} are to {2:graphs} whereas {2:states} are to {1:state diagrams}.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1734999643328-->
+END%%
+
 ## Determinism
 
 A **deterministic finite automaton** (DFA) is a $5$-tuple $\langle Q, \Sigma, \delta, q_0, F \rangle$, where
@@ -315,6 +329,31 @@ Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third ed
 <!--ID: 1734999643291-->
 END%%
 
+%%ANKI
+Basic
+Let $M$ be a DFA. What labels are permitted over arrows in its state diagram?
+Back: Members of its alphabet.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769438-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be a DFA. How many edges must leave a given state?
+Back: One for each symbol in its alphabet.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769445-->
+END%%
+
+%%ANKI
+Basic
+Is the following state diagram that of an NFA or DFA?
+![[dfa-example.png]]
+Back: Indeterminate.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769448-->
+END%%
+
 %%ANKI
 Basic
 Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be the depicted DFA. What does $Q$ evaluate to?
@@ -377,20 +416,6 @@ Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third ed
 <!--ID: 1734999643321-->
 END%%
 
-%%ANKI
-Cloze
-{1:Edges} are to {2:graphs} whereas {2:transitions} are to {1:state diagrams}.
-Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
-<!--ID: 1734999643325-->
-END%%
-
-%%ANKI
-Cloze
-{1:Vertices} are to {2:graphs} whereas {2:states} are to {1:state diagrams}.
-Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
-<!--ID: 1734999643328-->
-END%%
-
 %%ANKI
 Cloze
 The {final} states of a DFA are also called the {accept} states.
@@ -524,6 +549,329 @@ Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third ed
 <!--ID: 1734999643459-->
 END%%
 
+## Nondeterminism
+
+A **nondeterministic finite automaton** (NFA) is a $5$-tuple $\langle Q, \Sigma, \delta, q_0, F \rangle$, where
+
+1. $Q$ is a finite set called the **states**;
+2. $\Sigma$ is a finite set called the alphabet;
+3. $\delta: Q \times (\Sigma \cup \{\epsilon\}) \rightarrow \mathscr{P}(Q)$ is the **transition function**;
+4. $q_0 \in Q$ is the **start state**; and
+5. $F \subseteq Q$ is the set of **final states**.
+
+Like DFAs, these automaton are typically denoted using a **state diagram**. Unlike DFAs, not every state needs an exiting transition arrow for each symbol in the alphabet. Also, arrows can be labeled $\epsilon$ for the empty string.
+
+%%ANKI
+Basic
+A nondeterministic finite automaton is defined as a tuple of how many components?
+Back: Five.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769451-->
+END%%
+
+%%ANKI
+Basic
+What is NFA an acronym for?
+Back: **N**ondeterministic **f**inite **a**utomata.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769454-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What kind of mathematical entity is $Q$?
+Back: A finite set of states.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769457-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What name is given to $Q$?
+Back: $M$'s states.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769460-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What is $\Sigma$?
+Back: An alphabet.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769463-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What kind of mathematical entity is $\delta$?
+Back: A function.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769465-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What name is given to $\delta$?
+Back: $M$'s transition function.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769468-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What is $\delta$'s domain?
+Back: $Q \times (\Sigma \cup \{\epsilon\})$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769471-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What is $\delta$'s codomain?
+Back: $\mathscr{P}(Q)$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769474-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What kind of mathematical entity is $q_0$?
+Back: An urelement.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769477-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What name is given to $q_0$?
+Back: $M$'s start state.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769480-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What name is given to $F$?
+Back: $M$'s final states.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769483-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. What kind of mathematical entity is $F$?
+Back: A finite set.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769486-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. How does $F$ relate to $Q$?
+Back: $F \subseteq Q$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769490-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. How does $q_0$ relate to $Q$?
+Back: $q_0 \in Q$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769493-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be an NFA. How does $q_0$ relate to $F$?
+Back: N/A.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769496-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be an NFA. How many start states does $M$ have?
+Back: One.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769500-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be an NFA. How many accept states does $M$ have?
+Back: Zero or more.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769503-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be an NFA. How is $M$'s start state denoted in a state diagram?
+Back: With an arrow pointing to it from nowhere.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769506-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be an NFA. How is $M$'s final states denoted in a state diagram?
+Back: With double circles.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769510-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be an NFA. How is $M$'s transition function denoted in a state diagram?
+Back: As edges to and from states.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769513-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be an NFA. How is $M$'s alphabet denoted in a state diagram?
+Back: With symbols labeling each edge.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769517-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be an NFA. What labels are permitted over arrows in its state diagram?
+Back: Members of its alphabet or $\epsilon$.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769520-->
+END%%
+
+%%ANKI
+Basic
+Let $M$ be an NFA. How many edges must leave a given state?
+Back: Zero or more.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769523-->
+END%%
+
+%%ANKI
+Basic
+Is the following state diagram that of an NFA or DFA?
+![[nfa-example.png]]
+Back: NFA.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769525-->
+END%%
+
+%%ANKI
+Basic
+What two reasons explain why the following state diagram depicts an NFA?
+![[nfa-example.png]]
+Back: Missing labels/edges and existence of an $\epsilon$-labeled edge.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769528-->
+END%%
+
+%%ANKI
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be the depicted NFA. What does $Q$ evaluate to?
+![[nfa-example.png]]
+Back: $Q = \{q_1, q_2, q_3\}$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+END%%
+
+%%ANKI
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be the depicted NFA. What does $\Sigma$ evaluate to?
+![[nfa-example.png]]
+Back: $\Sigma = \{a, b\}$ or $\Sigma = \{a, b, \epsilon\}$.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be the depicted NFA. What does $q_0$ evaluate to?
+![[nfa-example.png]]
+Back: $q_0 = q_1$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769531-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be the depicted NFA. What does $\mathop{\text{dom}}\delta$ evaluate to?
+![[nfa-example.png]]
+Back: $\{q_1, q_2, q_3\} \times \{a, b, \epsilon\}$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769534-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be the depicted NFA. What does $\mathop{\text{ran}}\delta$ evaluate to?
+![[nfa-example.png]]
+Back: $\mathscr{P}(\{q_1, q_2, q_3\})$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769536-->
+END%%
+
+%%ANKI
+Basic
+Let $M = \langle Q, \Sigma, \delta, q_0, F \rangle$ be the depicted NFA. What does $F$ evaluate to?
+![[nfa-example.png]]
+Back: $\{q_1\}$
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769539-->
+END%%
+
+%%ANKI
+Basic
+What name is given to an NFA's standard graphical depiction?
+Back: Its state diagram.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769541-->
+END%%
+
+%%ANKI
+Cloze
+The {final} states of an NFA are also called the {accept} states.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769544-->
+END%%
+
+%%ANKI
+Basic
+Does the following NFA accept string `baba`?
+![[nfa-example.png]]
+Back: Yes.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769547-->
+END%%
+
+%%ANKI
+Basic
+Does the following NFA accept string `abab`?
+![[nfa-example.png]]
+Back: No.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769550-->
+END%%
+
+%%ANKI
+Basic
+Does the following NFA accept string `abba`?
+![[nfa-example.png]]
+Back: Yes.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769553-->
+END%%
+
+%%ANKI
+Basic
+Does the following NFA accept string `baab`?
+![[nfa-example.png]]
+Back: No.
+Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
+<!--ID: 1736781769556-->
+END%%
+
 ## Regular Operations
 
 Let $A$ and $B$ be languages. Then the **regular operations** union, intersection, concatenation, and Kleene star are defined as:

notes/set/cardinality.md

@@ -1482,7 +1482,7 @@ END%%
 
 %%ANKI
 Basic
-What does it mean for cardinal numbers to obey transitivity?
+What does it mean for cardinal number ordering to obey transitivity?
 Back: Let $\kappa$, $\lambda$, and $\mu$ be cardinal numbers. If $\kappa \leq \lambda$ and $\lambda \leq \mu$, then $\kappa \leq \mu$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1736702905252-->
@@ -1560,7 +1560,7 @@ 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: $A - \mathop{\text{ran}}g$
+Back: Either $A - \mathop{\text{ran}}g$ or $B - \mathop{\text{ran}} f$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1736711693546-->
 END%%