Standard graph representations, absolute value, triangle inequality.

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

@@ -197,7 +197,12 @@
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+    "infinite-cartesian-product.png",
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+    "algebra/abs-val.md": "a47bc08db62304eb526d15ede3e300cf",
+    "data-structures/graphs.md": "594d136ce637448641631c3647599c3a"
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notes/_journal/2024-03/2024-03-18.md

@@ -11,6 +11,6 @@ title: "2024-03-18"
 - [ ] Log Work Hours (Max 3 hours)
 
 * Finished [buffer pool manager project](https://15445.courses.cs.cmu.edu/fall2022/project1/).
-* Added notes on [[graphs|graph]]-related terminology. Updated "Introduction to Algorithms" to fourth edition.
+* Added notes on [[set/graphs|graph]]-related terminology. Updated "Introduction to Algorithms" to fourth edition.
 * Watched [How My Student Became 1 Dan](https://www.youtube.com/watch?v=ZvHL_lwfYYI&t=595s).
 * Reviewed and commented on Gus's latest pass of soft skills course.

notes/_journal/2024-05/2024-05-13.md

@@ -8,7 +8,7 @@ title: "2024-05-13"
 - [ ] Go (1 Life & Death Problem)
 - [ ] Korean (Read 1 Story)
 
-* Notes on [[graphs#Subgraphs|subgraphs]] and induced subgraphs.
+* Notes on [[set/graphs#Subgraphs|subgraphs]] and induced subgraphs.
 * Notes on [[remotes]].
 * Read through chapter 7 of "The Science of Programming", touching on the $wp$ predicate transformer.
 * Read chapter 1 of "Elements of Set Theory". Made some progress on chapter 2 which touches on the basic axiomatic foundations.

notes/_journal/2024-08-25.md

@@ -7,3 +7,6 @@ title: "2024-08-25"
 - [x] OGS
 - [ ] Sheet Music (10 min.)
 - [ ] Korean (Read 1 Story)
+
+* Standard [[data-structures/graphs|graph]] representations.
+* [[abs-val|Absolute value]] and the [[abs-val#Triangle Inequality|triangle inequality]].

notes/algebra/abs-val.md

@@ -0,0 +1,160 @@
+---
+title: Absolute Value
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algebra::abs
+tags:
+  - algebra
+---
+
+## Overview
+
+Let $x \in \mathbb{R}$. The **absolute value** of $x$, denoted $\lvert x \rvert$, is defined as $$\lvert x \rvert = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x \leq 0 \end{cases}$$
+
+%%ANKI
+Basic
+How is the absolute value of $x \in \mathbb{R}$ denoted?
+Back: $\lvert x \rvert$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565708-->
+END%%
+
+%%ANKI
+Basic
+How is the absolute value of $x \in \mathbb{R}$ defined?
+Back: $\lvert x \rvert = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x \leq 0 \end{cases}$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565711-->
+END%%
+
+%%ANKI
+Basic
+The absolute value of $x \in \mathbb{R}$ considers what two cases?
+Back: Whether $x \geq 0$ or $x \leq 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565713-->
+END%%
+
+%%ANKI
+Basic
+Let $x \in \mathbb{R}$. When is $-\lvert x \rvert \leq x < \lvert x \rvert$?
+Back: When $x < 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565715-->
+END%%
+
+%%ANKI
+Basic
+Let $x \in \mathbb{R}$. When is $-\lvert x \rvert < x \leq \lvert x \rvert$?
+Back: When $x > 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565716-->
+END%%
+
+%%ANKI
+Basic
+Let $x \in \mathbb{R}$. When is $-\lvert x \rvert \leq x \leq \lvert x \rvert$?
+Back: Always.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565717-->
+END%%
+
+%%ANKI
+Basic
+Let $x, a \in \mathbb{R}$ and $a \geq 0$. How is $\lvert x \rvert \leq a$ equivalently written as a chain of inequalities?
+Back: $-a \leq x \leq a$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565718-->
+END%%
+
+%%ANKI
+Basic
+Let $x, a \in \mathbb{R}$ and $a \geq 0$. How is $\lvert x \rvert \leq a$ geometricaly depicted?
+Back:
+![[abs-value-geom.png]]
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565719-->
+END%%
+
+%%ANKI
+Basic
+Let $x, a \in \mathbb{R}$ and $a \geq 0$. How is $-a \leq x \leq a$ equivalently written using absolute value?
+Back: $\lvert x \rvert \leq a$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565720-->
+END%%
+
+%%ANKI
+Basic
+Let $x, a \in \mathbb{R}$ and $a \geq 0$. How is $-a \leq x \leq a$ geometrically depicted?
+Back:
+![[abs-value-geom.png]]
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565721-->
+END%%
+
+## Triangle Inequality
+
+Let $x, y \in \mathbb{R}$. Then the **triangle inequality** of $\mathbb{R}$ states $$\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert$$
+
+%%ANKI
+Basic
+What does the triangle inequality of $\mathbb{R}$ state?
+Back: For $x, y \in \mathbb{R}$, $\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565722-->
+END%%
+
+%%ANKI
+Basic
+Why is the triangle inequality named the way it is?
+Back: The length of a triangle side is $\leq$ the sum of the other two sides.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724609565723-->
+END%%
+
+%%ANKI
+Basic
+What algebraic inequality is demonstrated in the following?
+![[triangle-inequality.png]]
+Back: The triangle inequality of $\mathbb{R}$.
+Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
+<!--ID: 1724609565724-->
+END%%
+
+%%ANKI
+Basic
+What degenerate triangle justifies use of $\leq$ over $<$ in the triangle inequality of $\mathbb{R}$?
+Back:
+![[triangle-inequality-degenerate.png]]
+Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
+<!--ID: 1724609565725-->
+END%%
+
+%%ANKI
+Basic
+What two chains of inequalities can be added together to prove the triangle inequality of $\mathbb{R}$?
+Back: $-\lvert x \rvert \leq x \leq \lvert x \rvert$ and $-\lvert y \rvert \leq y \leq \lvert y \rvert$.
+Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
+<!--ID: 1724609565726-->
+END%%
+
+%%ANKI
+Basic
+What does the general triangle inequality of $\mathbb{R}$ state?
+Back: For real numbers $a_1, \ldots, a_n$, $$\left\lvert \sum_{k=1}^n a_k \right\rvert \leq \sum_{k=1}^n \lvert a_k \rvert$$
+Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
+<!--ID: 1724611618744-->
+END%%
+
+%%ANKI
+Basic
+Let $a_1\, \ldots, a_n \in \mathbb{R}$. What is the following a generalization of? $$\left\lvert \sum_{k=1}^n a_k \right\rvert \leq \sum_{k=1}^n \lvert a_k \rvert$$
+Back: The triangle inequality of $\mathbb{R}$.
+Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
+<!--ID: 1724611618749-->
+END%%
+
+## Bibliography
+
+* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+* “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).

notes/data-structures/graphs.md

@@ -0,0 +1,194 @@
+---
+title: Graphs
+TARGET DECK: Obsidian::STEM
+FILE TAGS: data_structure::graph
+tags:
+  - data_structure
+  - graph
+---
+
+## Overview
+
+There are two standard ways of representing graphs in memory: **adjacency-list** representations and **adjacency-matrix** representations.
+
+%%ANKI
+Basic
+Using asymptotic notation, how do the number of edges in a graph relate to the number of vertices?
+Back: $\lvert E \rvert = O(\lvert V^2 \rvert)$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579417-->
+END%%
+
+%%ANKI
+Basic
+For graph $G = \langle V, E \rangle$, *why* is $\lvert E \rvert = O(\lvert V^2 \rvert)$?
+Back: Because $E$ is a binary relation on $V$.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579420-->
+END%%
+
+%%ANKI
+Basic
+What are the two standard ways of representing graphs in memory?
+Back: The adjacency-list and adjacency-matrix representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579422-->
+END%%
+
+%%ANKI
+Basic
+Which standard graph representation is preferred for sparse graphs?
+Back: Adjacency-list representations.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579423-->
+END%%
+
+%%ANKI
+Basic
+Which standard graph representation is preferred for dense graphs?
+Back: Adjacency-matrix representations.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579424-->
+END%%
+
+%%ANKI
+Basic
+When is a graph $G = \langle V, E \rangle$ considered dense?
+Back: When $\lvert E \rvert \approx \lvert V \rvert^2$.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579425-->
+END%%
+
+## Adjacency-List
+
+Let $G = \langle V, E \rangle$ be a graph. An adjacency-list representation of $G$ has an array of size $\lvert V \rvert$. Given $v \in V$, the index corresponding to $v$ contains a linked list containing all adjacent vertices.
+
+%%ANKI
+Basic
+Let $G = \langle V, E \rangle$ be a graph. It's adjacency-list representation is an array of what size?
+Back: $\lvert V \rvert$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579426-->
+END%%
+
+%%ANKI
+Basic
+The following is an example of what kind of graph representation?
+![[adj-list-representation.png]]
+Back: An adjacency-list representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579427-->
+END%%
+
+%%ANKI
+Basic
+Are adjacency-list representations used for directed or undirected graphs?
+Back: Both.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579428-->
+END%%
+
+%%ANKI
+Basic
+Let $G = \langle V, E \rangle$ be a graph. What is the sum of its adjacency-list representation's list lengths?
+Back: N/A. This depends on whether $G$ is a directed or undirected graph.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579429-->
+END%%
+
+%%ANKI
+Basic
+Let $G = \langle V, E \rangle$ be a digraph. What is the sum of its adjacency-list representation's list lengths?
+Back: $\lvert E \rvert$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579431-->
+END%%
+
+%%ANKI
+Basic
+Let $G = \langle V, E \rangle$ be an undirected graph. What is the sum of its adjacency-list representation's list lengths?
+Back: $2\lvert E \rvert$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579432-->
+END%%
+
+%%ANKI
+Basic
+Which lemma explains the sum of an undirected graph adjacency-list representation's list lengths?
+Back: The handshake lemma.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579433-->
+END%%
+
+%%ANKI
+Basic
+Let $G = \langle V, E \rangle$. What is the memory usage of its adjacency-list representation?
+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: 1724614579434-->
+END%%
+
+## Adjacency-Matrix
+
+Let $G = \langle V, E \rangle$ be a graph. An adjacency-matrix representation of $G$ is a $\lvert V \rvert \times \lvert V \rvert$ matrix $A = (a_{ij})$ such that $$a_{ij} = \begin{cases} 1 & \text{if } \langle i, j \rangle \in E \\ 0 & \text{otherwise} \end{cases}$$
+
+%%ANKI
+Basic
+Let $G = \langle V, E \rangle$ be a graph. It's adjacency-matrix representation is a matrix of what dimensions?
+Back: $\lvert V \rvert \times \lvert V \rvert$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579435-->
+END%%
+
+%%ANKI
+Basic
+What values are found in an adjacency-matrix representation of a graph?
+Back: $0$ and/or $1$.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579436-->
+END%%
+
+%%ANKI
+Basic
+The following is an example of what kind of graph representation?
+![[adj-matrix-representation.png]]
+Back: An adjacency-matrix representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579437-->
+END%%
+
+%%ANKI
+Basic
+Are adjacency-matrix representations used for directed or undirected graphs?
+Back: Both.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579438-->
+END%%
+
+%%ANKI
+Basic
+For what graphs are adjacency-matrix representations symmetric along its diagonal?
+Back: Undirected graphs.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579439-->
+END%%
+
+%%ANKI
+Basic
+*Why* is the adjacency-matrix representation of undirected graph $G = \langle V, E \rangle$ symmetric along its diagonal?
+Back: If $\langle i, j \rangle \in E$ then $\langle j, i \rangle \in E$.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579440-->
+END%%
+
+%%ANKI
+Basic
+Let $G = \langle V, E \rangle$. What is the memory usage of its adjacency-matrix representation?
+Back: $\Theta(\lvert V \rvert^2)$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1724614579441-->
+END%%
+
+## Bibliography
+
+* Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/set/natural-numbers.md

@@ -257,6 +257,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1724486757001-->
 END%%
 
+%%ANKI
+Basic
+In set theory, $\omega$ denotes what set?
+Back: The natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314391-->
+END%%
+
 %%ANKI
 Basic
 What is the smallest inductive set?
@@ -281,6 +289,86 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1724486757010-->
 END%%
 
+%%ANKI
+Basic
+What can be said about a subset of $\omega$?
+Back: N/A.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314394-->
+END%%
+
+%%ANKI
+Basic
+What can be said about an inductive subset of $\omega$?
+Back: It must coincide with $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314396-->
+END%%
+
+%%ANKI
+Basic
+Why must every inductive subset of $\omega$ coincide with $\omega$?
+Back: Because $\omega$ is the smallest inductive set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314397-->
+END%%
+
+%%ANKI
+Basic
+What does the induction principle for $\omega$ state?
+Back: Every inductive subset of $\omega$ coincides with $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314399-->
+END%%
+
+%%ANKI
+Basic
+What name is given to the principle, "every inductive subset of $\omega$ coincides with $\omega$?"
+Back: The induction principle for $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314400-->
+END%%
+
+%%ANKI
+Basic
+Inductive sets correspond to what kind of proof method?
+Back: Proof by induction.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314401-->
+END%%
+
+%%ANKI
+Basic
+Prove $P(n)$ is true for all $n \in \mathbb{N}$ using induction. What set do we prove is inductive?
+Back: $\{n \in \mathbb{N} \mid P(n)\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314403-->
+END%%
+
+%%ANKI
+Basic
+*How* are inductive sets and proof by induction related?
+Back: An induction proof corresponds to proving a related set is inductive.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314404-->
+END%%
+
+%%ANKI
+Basic
+What inductive set do we construct to prove the following by induction? $$\text{Every natural number is nonnegative}$$
+Back: $\{n \in \omega \mid 0 \leq n\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314405-->
+END%%
+
+%%ANKI
+Basic
+What inductive set do we construct to prove the following by induction? $$\text{Every nonzero natural number is the successor of another natural number}$$
+Back: $\{n \in \omega \mid n = 0 \lor (\exists m \in \omega, n = m^+)\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724606314406-->
+END%%
+
 ## Bibliography
 
 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

notes/set/trees.md

@@ -10,7 +10,7 @@ tags:
 
 ## Overview
 
-A **free tree** is a connected, acyclic, undirected [[graphs|graph]]. If an undirected graph is acyclic but possibly disconnected, it is a **forest**.
+A **free tree** is a connected, acyclic, undirected [[set/graphs|graph]]. If an undirected graph is acyclic but possibly disconnected, it is a **forest**.
 
 %%ANKI
 Basic
@@ -126,7 +126,7 @@ END%%
 
 A **rooted tree** is a free tree in which one vertex is distinguished/blessed as the **root**. We call vertices of rooted trees **nodes**.
 
-Let $T$ be a rooted tree with root $r$. Any node $y$ on the [[graphs#Paths|path]] from $r$ to node $x$ is an **ancestor** of $x$. Likewise, $x$ is a **descendant** of $y$. If the last edge on the path from $r$ to $x$ is $\{y, x\}$, $y$ is the **parent** of $x$ and $x$ is a **child** of $y$. Nodes with the same parent are called **siblings**.
+Let $T$ be a rooted tree with root $r$. Any node $y$ on the [[set/graphs#Paths|path]] from $r$ to node $x$ is an **ancestor** of $x$. Likewise, $x$ is a **descendant** of $y$. If the last edge on the path from $r$ to $x$ is $\{y, x\}$, $y$ is the **parent** of $x$ and $x$ is a **child** of $y$. Nodes with the same parent are called **siblings**.
 
 A node with no children is an **external node** or **leaf**. A node with at least one child is an **internal node** or **nonleaf**. The number of children of a node is the **degree** of said node. The length of the path from the root to a node $x$ is the **depth** of $x$ in $T$. A **level** of a tree consists of all nodes at the same depth. The **height** of a node in a tree is the length of the longest path from the node to a leaf.