diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
index b15b548..c7b334a 100644
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
+++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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diff --git a/notes/_journal/2024/2024-12/2024-12-31.md b/notes/_journal/2024/2024-12/2024-12-31.md
new file mode 100644
index 0000000..e624099
--- /dev/null
+++ b/notes/_journal/2024/2024-12/2024-12-31.md
@@ -0,0 +1,9 @@
+---
+title: "2024-12-31"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
\ No newline at end of file
diff --git a/notes/_journal/2025-01/2025-01-01.md b/notes/_journal/2025-01/2025-01-01.md
new file mode 100644
index 0000000..dc89a78
--- /dev/null
+++ b/notes/_journal/2025-01/2025-01-01.md
@@ -0,0 +1,12 @@
+---
+title: "2025-01-01"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Expanded notes on the basic property of integrals.
+* Notes on DFS [[dfs#Parenthesis Theorem|parenthesis theorem]] and related consequences.
\ No newline at end of file
diff --git a/notes/algorithms/dfs.md b/notes/algorithms/dfs.md
index dbe6810..06dfd15 100644
--- a/notes/algorithms/dfs.md
+++ b/notes/algorithms/dfs.md
@@ -149,7 +149,7 @@ END%%
 %%ANKI
 Basic
 What does a gray vertex typically represent?
-Back: A vertex that is in the queue, i.e. the frontier discovery happens against.
+Back: A vertex that is in the stack, i.e. the frontier discovery happens against.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1735081276189-->
 END%%
@@ -197,7 +197,7 @@ END%%
 %%ANKI
 Basic
 Suppose vertex $v$ has discovery time $d$ and finish time $f$. When was $v$ colored white?
-Back: At timestamps $< f$.
+Back: At timestamps $< d$.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1735081276199-->
 END%%
@@ -212,12 +212,23 @@ END%%
 
 %%ANKI
 Basic
-Suppose vertex $v$ has discovery time $d$ and finish time $t$. When was $v$ colored black?
+Suppose vertex $v$ has discovery time $d$ and finish time $f$. When was $v$ colored black?
 Back: At timestamps $\geq f$.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1735081276201-->
 END%%
 
+## Parenthesis Theorem
+
+In any depth-first search of a graph, for any two vertices $u$ and $v$, exactly one of the following three conditions holds:
+
+1. The intervals $[u.d, u.f]$ and $[v.d, v.f]$ are disjoint.
+	* No ancestor-descendant relation exists between $u$ and $v$.
+2. The interval $[u.d, u.f]$ is contained entirely within $[v.d, v.f]$.
+	* $u$ is a descendant of $v$.
+3. The interval $[v.d, v.f]$ is contained entirely within $[u.d, u.f]$.
+	* $v$ is a descendant of $u$.
+
 %%ANKI
 Basic
 What aspect of DFS has parenthesis structure?
@@ -226,6 +237,86 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1735081276202-->
 END%%
 
+%%ANKI
+Basic
+Let $u$ be a vertex. What does $u.d$ refer to?
+Back: When vertex $u$ was discovered during DFS.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232417-->
+END%%
+
+%%ANKI
+Basic
+Let $u$ be a vertex. What does $u.f$ refer to?
+Back: When vertex $u$ was finished processing during DFS.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232423-->
+END%%
+
+%%ANKI
+Basic
+Given vertices $a$ and $b$, what does the parenthesis theorem state?
+Back: Either $[a.d, a.f]$ and $[b.d, b.f]$ are disjoint or one interval is contained entirely within the other.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232427-->
+END%%
+
+%%ANKI
+Basic
+If $a$ is a descendant of $b$, what can be said about their discovery and finish times?
+Back: $b.d < a.d < a.f < b.f$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232431-->
+END%%
+
+%%ANKI
+Basic
+If $b$ is a descendant of $a$, what can be said about their discovery and finish times?
+Back: $a.d < b.d < b.f < a.f$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232435-->
+END%%
+
+%%ANKI
+Basic
+If $a$ and $b$ have no ancestor-descendant relation, what can be said about their discovery and finish times?
+Back: $[a.d, a.f]$ and $[b.d, b.f]$ are disjoint.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232439-->
+END%%
+
+%%ANKI
+Basic
+Let $a$ and $b$ be vertices with timestamps satisfying $a.d < b.d < b.f < a.f$. What ancestor-descendant relation exists?
+Back: $b$ is a descendant of $a$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232443-->
+END%%
+
+%%ANKI
+Basic
+Let $a$ and $b$ be vertices with timestamps satisfying $a.d < b.d < a.f < b.f$. What ancestor-descendant relation exists?
+Back: N/A. This is an impossible chain of inequalities.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232447-->
+END%%
+
+%%ANKI
+Basic
+Let $a$ and $b$ be vertices with timestamps satisfying $b.d < a.d < b.f < a.f$. What ancestor-descendant relation exists?
+Back: N/A. This is an impossible chain of inequalities.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232451-->
+END%%
+
+%%ANKI
+Basic
+Let $a$ and $b$ be vertices with timestamps satisfying $b.d < a.d < a.f < b.f$. What ancestor-descendant relation exists?
+Back: $a$ is a descendant of $b$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232455-->
+END%%
+
 ## Depth-First Forests
 
 To color an entire graph black, BFS may need to be invoked multiple times. After each invocation of BFS, a new invocation can be run with any remaining white vertex as the source. Each invocation yields a **depth-first tree**. Multiple invocations yield a **depth-first forest**.
@@ -292,7 +383,7 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1735081276210-->
 END%%
 
-### Edge Classification
+## Edge Classification
 
 A depth-first forest can contain four different types of edges:
 
@@ -329,7 +420,7 @@ END%%
 %%ANKI
 Basic
 Let $\langle u, v \rangle$ be an edge in a depth-first forest. When is $\langle u, v \rangle$ a back edge?
-Back: When $v$ is an ancestor of edge $u$.
+Back: When $v$ is an ancestor of $u$.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1735085295263-->
 END%%
@@ -443,7 +534,6 @@ END%%
 
 %%ANKI
 Basic
-Assuming solid lines are tree edges, *why* is edge $\langle b, s \rangle$ a back edge?
 ![[dfs-edge-classification.png]]
 Back: Because $s$ is an ancestor of $b$.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
@@ -468,6 +558,62 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1735086202741-->
 END%%
 
+%%ANKI
+Basic
+Let $v$ be white when $\langle u, v \rangle$ is explored. What kind of edge is $\langle u, v \rangle$ classified as?
+Back: A tree edge.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232459-->
+END%%
+
+%%ANKI
+Basic
+Let $v$ be gray when $\langle u, v \rangle$ is explored. What kind of edge is $\langle u, v \rangle$ classified as?
+Back: A back edge.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232463-->
+END%%
+
+%%ANKI
+Basic
+Let $v$ be black when $\langle u, v \rangle$ is explored. What kind of edge is $\langle u, v \rangle$ classified as?
+Back: Either a forward edge or cross edge.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232469-->
+END%%
+
+%%ANKI
+Basic
+Let $v$ be black when $\langle u, v \rangle$ is explored. If $u.d < v.d$, what kind of edge is $\langle u, v \rangle$ classified as?
+Back: A forward edge.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232475-->
+END%%
+
+%%ANKI
+Basic
+Let $v$ be black when $\langle u, v \rangle$ is explored. If $v.d < u.d$, what kind of edge is $\langle u, v \rangle$ classified as?
+Back: A cross edge.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785232480-->
+END%%
+
+%%ANKI
+Basic
+When performing DFS on a directed graph, what possible edge classifications are there?
+Back: Tree, forward, back, and cross.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785623431-->
+END%%
+
+%%ANKI
+Basic
+When performing DFS on an undirected graph, what possible edge classifications are there?
+Back: Tree and back.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1735785623439-->
+END%%
+
 ## Bibliography
 
 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
\ No newline at end of file
diff --git a/notes/c17/macros.md b/notes/c17/macros.md
index 1ab76a6..094abd7 100644
--- a/notes/c17/macros.md
+++ b/notes/c17/macros.md
@@ -308,7 +308,7 @@ Given the following translation unit, what is output to `stdout`?
 
 int main(void) { printf("%s\n", foo(123, 456)); }
 ```
-Back: N/A. Pasting formed `"123""456"`, an invalid preprocessing token.
+Back: N/A. Pasting forms `"123""456"`, an invalid preprocessing token.
 Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 <!--ID: 1733308613501-->
 END%%
@@ -321,7 +321,7 @@ Given the following translation unit, what is output to `stdout`?
 
 int main(void) { printf("%s\n", foo("123", "456")); }
 ```
-Back: N/A. Pasting formed `"123""456"`, an invalid preprocessing token.
+Back: N/A. Pasting forms `"123""456"`, an invalid preprocessing token.
 Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 <!--ID: 1733308613504-->
 END%%
diff --git a/notes/c17/storage.md b/notes/c17/storage.md
index ed29e36..cfdf6f5 100644
--- a/notes/c17/storage.md
+++ b/notes/c17/storage.md
@@ -238,7 +238,7 @@ END%%
 %%ANKI
 Basic
 What is the lifetime of an object with thread storage duration?
-Back: The entire execution fo the thread for which it is created.
+Back: The entire execution of the thread for which it is created.
 Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf).
 <!--ID: 1733060664939-->
 END%%
diff --git a/notes/calculus/integrals.md b/notes/calculus/integrals.md
index 40a354e..3b8b8eb 100644
--- a/notes/calculus/integrals.md
+++ b/notes/calculus/integrals.md
@@ -13,7 +13,7 @@ The integral is usually defined first in terms of step functions and then genera
 
 Suppose $f$ is [[#Integrable Functions|integrable]] on interval $[a, b]$. Then the **integral** of $f$ from $a$ to $b$ is denoted as $$\int_a^b f(x) \,dx.$$
 
-The **lower limit of integration** is $a$. The **upper limit of integration** is $b$. Together they form the **limits of integration**. $f(x)$ is called the **integrand** whereas $dx$ is called the **differential**. Furthermore, we define $$\int_a^b f(x) \,dx = -\int_b^a f(x) \,dx \quad\text{and}\quad \int_a^a f(x)\,dx = 0.$$
+The **lower limit of integration** is $a$. The **upper limit of integration** is $b$. Together they form the **integration limits**. $f(x)$ is called the **integrand** whereas $dx$ is called the **differential**. Furthermore, we define $$\int_a^b f(x) \,dx = -\int_b^a f(x) \,dx \quad\text{and}\quad \int_a^a f(x)\,dx = 0.$$
 
 %%ANKI
 Basic
@@ -49,7 +49,7 @@ END%%
 
 %%ANKI
 Basic
-Let $f$ be integrable over $[a, b]$. What does $\int_a^b f(x) \,dx$ evaluate to after swapping limits of integration?
+Let $f$ be integrable over $[a, b]$. What does $\int_a^b f(x) \,dx$ evaluate to after swapping integration limits?
 Back: $-\int_b^a f(x) \,dx$.
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1734816555515-->
@@ -90,7 +90,7 @@ END%%
 %%ANKI
 Basic
 What name is given collectively to $a$ and $b$ in $\int_a^b f(x) \,dx$?
-Back: The limits of integration.
+Back: The integration limits.
 Reference: “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875).
 <!--ID: 1734816555526-->
 END%%
@@ -269,27 +269,51 @@ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Int
 <!--ID: 1734814463673-->
 END%%
 
-### Homogeneousness
+### Vertical Scaling
 
-Let $f$ be integrable over $[a, b]$ and $c \in \mathbb{R}$. Then $$\int_a^b c \cdot f(x) \,dx = c\int_a^b f(x) \,dx$$
+Let $f$ be integrable over $[a, b]$ and $c \in \mathbb{R}$. Then $$\int_a^b c \cdot f(x) \,dx = c\int_a^b f(x) \,dx.$$
+
+This is also known as the **homogeneous property**.
 
 %%ANKI
 Basic
 What does the homogeneous property of integrals state?
-Back: Let $f$ be integrable over $[a, b]$ and $c \in \mathbb{R}$. Then $\int_a^b c \cdot f(x) \,dx = c \int_a^b f(x) \,dx$.
+Back: Let $f$ be integrable over $[a, b]$ and $c \in \mathbb{R}$. Then $$\int_a^b c \cdot f(x) \,dx = c \int_a^b f(x) \,dx.$$
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1734814463679-->
 END%%
 
+%%ANKI
+Basic
+What does the vertical scaling property of integrals state?
+Back: Let $f$ be integrable over $[a, b]$ and $c \in \mathbb{R}$. Then $$\int_a^b c \cdot f(x) \,dx = c \int_a^b f(x) \,dx.$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735744900572-->
+END%%
+
 %%ANKI
 Basic
 Assume the following integrals are defined. What is the following identity called? $$\int_a^b c \cdot f(x) \,dx = c\int_a^b f(x) \,dx$$
 
-Back: The homogeneous property.
+Back: The vertical scaling (or homogeneous) property.
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1734814463685-->
 END%%
 
+%%ANKI
+Cloze
+The {vertical scaling} property of integrals is also known as the {homogeneous} property.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735745285982-->
+END%%
+
+%%ANKI
+Cloze
+The {vertical scaling} property of integrals corresponds to a change of scale on the {$y$}-axis.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735743913546-->
+END%%
+
 ### Linearity
 
 Let $f$ and $g$ be integrable over $[a, b]$. Let $c_1, c_2 \in \mathbb{R}$. Then $$\int_a^b [c_1f(x) + c_2g(x)] \,dx = c_1 \int_a^b f(x) \,dx + c_2 \int_a^b g(x) \,dx$$
@@ -313,14 +337,14 @@ END%%
 %%ANKI
 Basic
 The linearity property is immediately derived from what other two properties?
-Back: The additive property w.r.t. the integrand and homogeneousness.
+Back: The additive property w.r.t. the integrand and vertical scaling.
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1734814463710-->
 END%%
 
 %%ANKI
 Cloze
-The {linearity} property of integrals is an immediate consequence of {additivity w.r.t the integrand} and {homogenousness}.
+The {linearity} property of integrals follows from the {additivity w.r.t the integrand} property and the {homogenous} property.
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1734814463693-->
 END%%
@@ -363,12 +387,12 @@ END%%
 
 ### Interval of Integration Additivity
 
-If two of the following three integrals exist, the third also exists, and we have $$\int_a^b f(x) \,dx + \int_b^c f(x) \,dx = \int_a^c f(x) \,dx$$
+Let $f$ be integrable over an interval containing $a$, $b$, and $c$. Then $$\int_a^b f(x) \,dx + \int_b^c f(x) \,dx = \int_a^c f(x) \,dx$$
 
 %%ANKI
 Basic
 What does the additivity property w.r.t. the interval of integration state?
-Back: If two of the following three integrals exist, the third also exists, and we have $$\int_a^b f(x) \,dx + \int_b^c f(x) \,dx = \int_a^c f(x) \,dx$$
+Back: Let $f$ be integrable over an interval containing $a$, $b$, and $c$. Then $$\int_a^b f(x) \,dx + \int_b^c f(x) \,dx = \int_a^c f(x) \,dx$$
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1735613867221-->
 END%%
@@ -458,6 +482,181 @@ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Int
 <!--ID: 1735616354222-->
 END%%
 
+%%ANKI
+Basic
+How is the following rewritten so that the integrand is just $f(x)$? $$\int_a^b f(x + c) \,dx$$
+Back: $$\int_{a+c}^{b+c} f(x) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735750191049-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten so that the integrand is just $f(x)$? $$\int_a^b f(x - c) \,dx$$
+Back: $$\int_{a-c}^{b-c} f(x) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735750317904-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten so that the integration limits are just $a$ and $b$? $$\int_{a+c}^{b+c} f(x) \,dx$$
+Back: $$\int_{a}^{b} f(x + c) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735750317913-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten so that the integration limits are just $a$ and $b$? $$\int_{a-c}^{b-c} f(x) \,dx$$
+Back: $$\int_{a}^{b} f(x - c) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735751159481-->
+END%%
+
+### Horizontal Scaling
+
+Let $f$ be integrable on $[a, b]$. Then for all $k \in \mathbb{R}$ such that $k \neq 0$, $$\int_a^b f(x) \,dx = \frac{1}{k} \int_{ka}^{kb} f\left(\frac{x}{k}\right) \,dx.$$
+
+%%ANKI
+Basic
+What does the horizontal scaling property of integrals state?
+Back: Let $f$ be integrable over $[a, b]$ and $k \in \mathbb{R}$ s.t. $k \neq 0$. Then $$\int_a^b f(x) \,dx = \frac{1}{k} \int_{ka}^{kb} f\left(\frac{x}{k}\right) \,dx.$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735743913556-->
+END%%
+
+%%ANKI
+Basic
+Assume the following integrals are defined. What is the following identity called? $$\int_a^b f(x) \,dx = \frac{1}{k} \int_{ka}^{kb} f\left(\frac{x}{k}\right) \,dx$$
+Back: The horizontal scaling property.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735743913568-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be integrable over $[a, b]$ and $g(x) = f(x / k)$. What integral of $g$ equals $\int_a^b f(x) \,dx$?
+Back: $\frac{1}{k} \int_{ka}^{kb} g(x) \,dx$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735743913575-->
+END%%
+
+%%ANKI
+Basic
+Let $f$ be integrable over $[a, b]$ and $g(x) = f(kx)$. What integral of $g$ equals $\int_a^b f(x) \,dx$?
+Back: $k \int_{a/k}^{b/k} g(x) \,dx$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735743913580-->
+END%%
+
+%%ANKI
+Cloze
+The {horizontal scaling} property of integrals corresponds to a change of scale on the {$x$}-axis.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735743913591-->
+END%%
+
+%%ANKI
+Basic
+What identity relates horizontal and vertical scaling of integrals?
+Back: $$\int_a^b kf(x) \,dx = \int_{ka}^{kb} f\left(\frac{x}{k}\right) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735749673647-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten so that the integrand is just $f(x)$? $$\int_a^b f(kx) \,dx$$
+Back: $$\frac{1}{k} \int_{ka}^{kb} f(x) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735750191055-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten so that the integrand is just $f(x)$? $$\int_a^b f\left(\frac{x}{k}\right) \,dx$$
+Back: $$k \int_{a / k}^{b / k} f(x) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735751159485-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten so that the integration limits are just $a$ and $b$? $$\int_{ka}^{kb} f(x) \,dx$$
+Back: $$k \int_{a}^{b} f(kx) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735751159489-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten so that the integration limits are just $a$ and $b$? $$\int_{a/k}^{b/k} f(x) \,dx$$
+Back: $$\frac{1}{k} \int_{a}^{b} f\left(\frac{x}{k}\right) \,dx$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735751159493-->
+END%%
+
+### Reflection
+
+The reflection properties are special cases of the vertical and horizontal scaling properties. Let $f$ be integrable over $[a, b]$. Then $$\int_a^b f(x) \,dx = -\int_a^b -f(x) \,dx \quad\text{and}\quad \int_a^b f(x) \,dx = \int_{-b}^{-a} f(-x) \,dx.$$
+
+%%ANKI
+Basic
+What does the horizontal reflection property of integrals state?
+Back: Let $f$ be integrable over $[a, b]$. Then $$\int_a^b f(x) \,dx = \int_{-b}^{-a} f(-x) \,dx.$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735744900576-->
+END%%
+
+%%ANKI
+Basic
+What does the vertical reflection property of integrals state?
+Back: Let $f$ be integrable over $[a, b]$. Then $$\int_a^b f(x) \,dx = -\int_{a}^{b} -f(x) \,dx.$$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735745285987-->
+END%%
+
+%%ANKI
+Basic
+Assume the following integrals are defined. What is the following identity called? $$\int_a^b f(x) \,dx = \int_{-b}^{-a} f(-x) \,dx.$$
+Back: The horizontal reflection property.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735744900581-->
+END%%
+
+%%ANKI
+Basic
+Assume the following integrals are defined. What is the following identity called? $$\int_a^b f(x) \,dx = -\int_{a}^{b} -f(x) \,dx.$$
+Back: The vertical reflection property.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735745285989-->
+END%%
+
+%%ANKI
+Basic
+The horizontal reflection property of integrals is a special case of what other basic property?
+Back: Horizontal scaling.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735744900585-->
+END%%
+
+%%ANKI
+Basic
+The vertical reflection property of integrals is a special case of what other basic property?
+Back: Vertical scaling.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735745285992-->
+END%%
+
+%%ANKI
+Basic
+The reflection property of integrals is a scaling by what value?
+Back: $-1$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1735744900590-->
+END%%
+
 ## Bibliography
 
 * “Integral.” In _Wikipedia_, December 31, 2024. [https://en.wikipedia.org/w/index.php?title=Integral](https://en.wikipedia.org/w/index.php?title=Integral&oldid=1266307875).
diff --git a/notes/computability/automaton.md b/notes/computability/automaton.md
index bb1e07b..db023bc 100644
--- a/notes/computability/automaton.md
+++ b/notes/computability/automaton.md
@@ -25,7 +25,7 @@ A [[language]] is called a **regular language** if a finite automaton recognizes
 
 %%ANKI
 Basic
-A finite automaton comprises of how many components?
+A 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: 1734999643206-->
@@ -130,7 +130,7 @@ END%%
 %%ANKI
 Basic
 Consider finite automaton $M = \langle Q, \Sigma, \delta, q_0, F \rangle$. How does $q_0$ relate to $Q$?
-Back: $q_0 \subseteq Q$
+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: 1734999643257-->
 END%%
@@ -255,7 +255,7 @@ END%%
 
 %%ANKI
 Cloze
-{1:Edges} are to {2:diagraphs} whereas {2:transitions} are to {1:state diagrams}.
+{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%%
@@ -540,7 +540,7 @@ END%%
 %%ANKI
 Basic
 Let $A$ be a language. How is the Kleene star regular operation defined?
-Back: As $A^* = \{ x_1x_2 \cdots x_k \mid k \geq 0 \land x_i \in A \}$
+Back: As $A^* = \{ x_1x_2 \cdots x_k \mid k \geq 0 \land x_1, \ldots, x_k \in A \}$
 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: 1735160593075-->
 END%%
diff --git a/notes/geometry/cartesian.md b/notes/geometry/cartesian.md
index 842cb79..729977a 100644
--- a/notes/geometry/cartesian.md
+++ b/notes/geometry/cartesian.md
@@ -723,6 +723,20 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces
 <!--ID: 1734620943953-->
 END%%
 
+%%ANKI
+Cloze
+A {horizontal} reflection mirrors over/through the {$y$}-axis.
+Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
+<!--ID: 1735744900558-->
+END%%
+
+%%ANKI
+Cloze
+A {vertical} reflection mirrors over/through the {$x$}-axis.
+Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
+<!--ID: 1735744900564-->
+END%%
+
 %%ANKI
 Basic
 Consider the graph of $f(x)$. Is $f(-x)$ its horizontal or vertical reflection?
@@ -771,6 +785,14 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces
 <!--ID: 1734656938178-->
 END%%
 
+%%ANKI
+Basic
+A reflection of a graph is a scaling by what value?
+Back: $-1$
+Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
+<!--ID: 1735744900567-->
+END%%
+
 ## Bibliography
 
 * “Cartesian Coordinate System,” in _Wikipedia_, October 21, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system](https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system&oldid=1252434514).
diff --git a/notes/linkers/relocatable.md b/notes/linkers/relocatable.md
index bf445ac..ece3aa7 100644
--- a/notes/linkers/relocatable.md
+++ b/notes/linkers/relocatable.md
@@ -118,7 +118,7 @@ END%%
 %%ANKI
 Basic
 With respect to linkers, a local symbol corresponds to what kind of C construct?
-Back: A static function or global variable defined in the given module.
+Back: A static function or variable defined in the given module.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: c17
 <!--ID: 1735159138677-->
diff --git a/notes/set/functions.md b/notes/set/functions.md
index faa04c6..fe4ea87 100644
--- a/notes/set/functions.md
+++ b/notes/set/functions.md
@@ -255,7 +255,7 @@ END%%
 %%ANKI
 Basic
 How is a permutation defined using the concept of functions?
-Back: A permutation of a set $A$ is a one-to-one function from $A$ to $A$.
+Back: A permutation of a set $A$ is a one-to-one function from $A$ onto $A$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1735074143690-->
 END%%
diff --git a/notes/set/graphs.md b/notes/set/graphs.md
index 7f55954..26e4312 100644
--- a/notes/set/graphs.md
+++ b/notes/set/graphs.md
@@ -632,7 +632,7 @@ END%%
 Basic
 Which vertices are isolated in the following?
 ![[directed-graph-example.png]]
-Back: N/A
+Back: N/A.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1710796091112-->
 END%%