Additive and multiplicative principles.

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notes/_journal/2024-02-17.md

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 - [x] Anki Flashcards
 - [x] KoL
 - [ ] Sheet Music (10 min.)
-- [ ] OGS (1 Life & Death Problem)
+- [x] OGS (1 Life & Death Problem)
-- [ ] Korean (Read 1 Story)
+- [x] Korean (Read 1 Story)
 - [ ] Interview Prep (1 Practice Problem)
 - [ ] Log Work Hours (Max 3 hours)
 
 * Began working through two's-complement and unsigned encoding.
 * Also added more notes on propositional logic.
 	* Taking a step back from "Concrete Mathematics" to first read through "Discrete Mathematics: An Open Introduction".
+* 101weiqi problems (serial numbers)
+	* Q-14584
+	* Q-253960
+	* Q-21834
+	* Q-349936
+	* Q-9812
+* Read 금도끼와 은도끼 (The Golden Ax and the Silver Ax).
+* Read through first section of "Discrete Mathematics: An Open Introduction". Add combinatorics notes.

notes/combinatorics/index.md

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+---
+title: Combinatorics
+TARGET DECK: Obsidian::STEM
+FILE TAGS: combinatorics set
+tags:
+  - combinatorics
+  - set
+---
+
+## Overview
+
+The **additive principle** states that two finite and disjoint sets $A$ and $B$ satisfy $$|A \cup B| = |A| + |B|$$
+
+The **multiplicative principle** states that two finite sets $A$ and $B$ satisfy $$|A \times B| = |A| \cdot |B|$$
+
+%%ANKI
+Basic
+What does the additive principle state?
+Back: Given finite and disjoint sets $A$ and $B$, $|A \cup B| = |A| + |B|$.
+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: 1708217738464-->
+END%%
+
+%%ANKI
+Basic
+What does the multiplicative principle state?
+Back: Given finite sets $A$ and $B$, $|A \times B| = |A| \cdot |B|$.
+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: 1708217738469-->
+END%%
+
+%%ANKI
+Basic
+The additive property applies to sets exhibiting what two properties?
+Back: Finiteness and disjointedness.
+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: 1708217738473-->
+END%%
+
+%%ANKI
+Basic
+The multiplicative property applies to sets exhibiting what property?
+Back: Finiteness.
+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: 1708217738477-->
+END%%
+
+%%ANKI
+Cloze
+The additive principle is to {$\cup$} whereas the multiplicative principle is to {$\times$}.
+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: 1708217738480-->
+END%%
+
+%%ANKI
+Basic
+If $A$ is finite, how is $A \times B$ rewritten as $|A|$ disjoint sets?
+Back: Given $A = \{a_1, \ldots, a_n\}$, $(\{a_1\} \times B) \cup \cdots \cup (\{a_n\} \times B)$.
+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: 1708217738483-->
+END%%
+
+%%ANKI
+Basic
+If $B$ is finite, how is $A \times B$ rewritten as $|B|$ disjoint sets?
+Back: Given $B = \{b_1, \ldots, b_n\}$, $(A \times \{b_1\}) \cup \cdots \cup (A \times \{b_n\})$.
+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: 1708217738487-->
+END%%
+
+%%ANKI
+Basic
+How do we denote $A$ and $B$ are disjoint using standard set notation?
+Back: $A \cap B = \varnothing$
+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: 1708217738491-->
+END%%
+
+%%ANKI
+Basic
+How is the cartesian product $A \times B$ defined?
+Back: $A \times B = \{\langle x, y \rangle : x \in A \land y \in B\}$
+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: 1708217738494-->
+END%%
+
+## References
+
+* 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).