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Additive and multiplicative principles.

c-declarations
Joshua Potter 2024-02-17 17:56:55 -07:00
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- [x] Anki Flashcards
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* 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".
* 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.

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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).