Notes on bags and multigraphs.

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notes/_journal/2024-07-07.md

@@ -0,0 +1,11 @@
+---
+title: "2024-07-07"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[bags]] and multigraphs.

notes/set/bags.md

@@ -0,0 +1,144 @@
+---
+title: Bags
+TARGET DECK: Obsidian::STEM
+FILE TAGS: set::bag
+tags:
+  - bag
+  - set
+---
+
+## Overview
+
+Sets can contain the same element only once. This means $\{1, 1\} = \{1\}$. A **bag** (or **multiset**) can include a single element multiple times.
+
+%%ANKI
+Basic
+What is a bag?
+Back: A set in which we are allowed to include a single element multiple times.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266774-->
+END%%
+
+%%ANKI
+Basic
+What alternative term do "bags" go by?
+Back: Multisets.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266800-->
+END%%
+
+%%ANKI
+Cloze
+The {multiplicity} of an element in a bag is the {number of instances given for that element}.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266806-->
+END%%
+
+%%ANKI
+Basic
+How many bags exist with members $a$ and $b$?
+Back: An infinite amount.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266813-->
+END%%
+
+%%ANKI
+Basic
+What is the multiplicity of $a$ in bag $\{a, b\}$?
+Back: $1$
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266819-->
+END%%
+
+%%ANKI
+Basic
+What is the multiplicity of $a$ in bag $\{a, a, a, b, b, c\}$?
+Back: $3$
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266825-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes a bag from a tuple?
+Back: Order is irrelevant in the former.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266830-->
+END%%
+
+%%ANKI
+Basic
+What is the cardinality of $a$ in bag $\{a, a, a, b, b, c\}$?
+Back: N/A. Cardinality applies to bags, not members.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266835-->
+END%%
+
+%%ANKI
+Basic
+What is the cardinality of bag $\{a, a, a, b, b, c\}$?
+Back: $6$
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266840-->
+END%%
+
+%%ANKI
+Basic
+How is the cardinality of a bag calculated?
+Back: By summing the multiplicities of each member.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266845-->
+END%%
+
+%%ANKI
+Basic
+When viewed as a function, a multiset has domain what?
+Back: The set comprising each member of the multiset.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266850-->
+END%%
+
+%%ANKI
+Basic
+When viewed as a function, a multiset has range what?
+Back: The set comprising the multiplicities of the multiset's members.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266856-->
+END%%
+
+%%ANKI
+Basic
+When viewed as a function, a multiset has codomain what?
+Back: The positive integers.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266861-->
+END%%
+
+%%ANKI
+Basic
+Do all multisets correspond to sets?
+Back: No.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266867-->
+END%%
+
+%%ANKI
+Basic
+Do all sets correspond to multisets?
+Back: Yes.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266873-->
+END%%
+
+%%ANKI
+Basic
+Under what condition is a bag also a set?
+Back: When the multiplicity of each element in the bag is $1$.
+Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+<!--ID: 1720360266878-->
+END%%
+
+## Bibliography
+
+* “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
+* 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).

notes/set/graphs.md

@@ -243,6 +243,49 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1710793937921-->
 END%%
 
+A graph that allows multiple edges between vertices is called a **multigraph**. It is analagous to the concept of [[bags|multisets]] in set theory.
+
+%%ANKI
+Basic
+What is a multigraph?
+Back: A graph with multiple edges between any two vertices.
+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: 1720360545669-->
+END%%
+
+%%ANKI
+Cloze
+{Multigraphs} are to graph theory as {multisets} are to set theory.
+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: 1720360545673-->
+END%%
+
+%%ANKI
+Basic
+Does every multigraph correspond to a graph?
+Back: No.
+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: 1720360545677-->
+END%%
+
+%%ANKI
+Basic
+Does every graph correspond to a multigraph?
+Back: Yes.
+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: 1720360545680-->
+END%%
+
+%%ANKI
+Basic
+Under what conditions is a multigraph considered a graph?
+Back: When the number of edges between any two vertices is $1$.
+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: 1720360545684-->
+END%%
+
+## Incidence
+
 If $\langle u, v \rangle$ is an edge of a directed graph, we say $\langle u, v \rangle$ is **incident to** $v$ and **incident from** $u$. Furthermore, we say $v$ is **adjacent** to $u$. If $\{u, v\}$ was instead an edge of an undirected graph, we say $\{u, v\}$ is **incident on** $u$ and $v$. Likewise, $v$ is adjacent to $u$ and $u$ is adjacent to $v$.
 
 %%ANKI
@@ -1120,7 +1163,7 @@ END%%
 Basic
 If the following graphs are isomorphic, what is the domain of the isomorphism?
 ![[graph-isomorphic.png]]
-Back: $\{a, b, c\}$.
+Back: $\{a, b, c\}$
 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: 1715537560210-->
 END%%
@@ -1129,7 +1172,7 @@ END%%
 Basic
 If the following graphs are isomorphic, what is the codomain of the isomorphism?
 ![[graph-isomorphic.png]]
-Back: $\{u, v, w\}$.
+Back: $\{u, v, w\}$
 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: 1715537560214-->
 END%%

notes/set/index.md

@@ -8,6 +8,8 @@ tags:
 
 ## Overview
 
+Set theory begins with two primitive notions of sets and membership. Other axioms are defined relative to these concepts.
+
 %%ANKI
 Basic
 What are the two primitive notions of set theory?