122 lines
5.2 KiB
Markdown
122 lines
5.2 KiB
Markdown
---
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title: Combinatorics
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TARGET DECK: Obsidian::STEM
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FILE TAGS: combinatorics set
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tags:
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- combinatorics
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- set
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---
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## Overview
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The **multiplicative principle** states that two finite sets $A$ and $B$ satisfy $$|A \times B| = |A| \cdot |B|$$
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This can be generalized to any number of finite sets in the obvious way.
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%%ANKI
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Basic
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What does the multiplicative principle state?
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Back: Given finite sets $A$ and $B$, $|A \times B| = |A| \cdot |B|$.
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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).
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<!--ID: 1708217738469-->
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END%%
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%%ANKI
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Basic
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The multiplicative property applies to sets exhibiting what property?
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Back: Finiteness.
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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).
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END%%
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%%ANKI
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Cloze
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{`union`} is to the additive property whereas {`struct`} is to the multiplicative property.
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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).
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Tags: c17
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<!--ID: 1708221293483-->
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END%%
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%%ANKI
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Basic
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Which C construct corresponds to the multiplicative property?
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Back: `struct`
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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).
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Tags: c17
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<!--ID: 1708221293489-->
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END%%
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%%ANKI
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Cloze
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The additive principle is to {$\cup$} whereas the multiplicative principle is to {$\times$}.
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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).
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<!--ID: 1708217738480-->
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END%%
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%%ANKI
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Basic
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If $A$ is finite, how is $A \times B$ rewritten as $|A|$ disjoint sets?
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Back: Given $A = \{a_1, \ldots, a_n\}$, $(\{a_1\} \times B) \cup \cdots \cup (\{a_n\} \times B)$.
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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).
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<!--ID: 1708217738483-->
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END%%
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%%ANKI
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Basic
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If $B$ is finite, how is $A \times B$ rewritten as $|B|$ disjoint sets?
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Back: Given $B = \{b_1, \ldots, b_n\}$, $(A \times \{b_1\}) \cup \cdots \cup (A \times \{b_n\})$.
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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).
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<!--ID: 1708217738487-->
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END%%
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%%ANKI
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Basic
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How is the cartesian product $A \times B$ defined?
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Back: $A \times B = \{\langle x, y \rangle : x \in A \land y \in B\}$
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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).
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<!--ID: 1708217738494-->
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END%%
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%%ANKI
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Basic
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How many functions exist between $\{1, 2, 3, 4, 5\}$ and $\{a, b, c, d\}$?
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Back: $4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4^5$
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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).
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<!--ID: 1708221293492-->
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END%%
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%%ANKI
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Basic
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How many functions exist between finite sets $A$ and $B$?
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Back: $|B|^{|A|}$
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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).
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<!--ID: 1708221293496-->
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END%%
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%%ANKI
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Basic
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What combinatorial concept explains the number of functions between two finite sets?
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Back: The multiplicative principle.
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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).
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<!--ID: 1708351482412-->
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END%%
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%%ANKI
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Basic
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How is the "count of three letter license plates" reimagined as a count of functions?
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Back: As the number of functions from $\{1, 2, 3\}$ to $\{A, B, \ldots, Z\}$.
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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).
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<!--ID: 1708221293499-->
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END%%
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%%ANKI
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Basic
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How is the "maximum unsigned $w$-bit number" reimagined as a count of functions?
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Back: As one less than the number of functions from $\{1, 2, \ldots, w\}$ to $\{0, 1\}$.
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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).
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<!--ID: 1708221293502-->
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END%%
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## References
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* 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). |