89 lines
3.8 KiB
Markdown
89 lines
3.8 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 **additive principle** states that two finite and disjoint sets $A$ and $B$ satisfy $$|A \cup B| = |A| + |B|$$
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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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%%ANKI
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Basic
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What does the additive principle state?
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Back: Given finite and disjoint sets $A$ and $B$, $|A \cup B| = |A| + |B|$.
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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: 1708217738464-->
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END%%
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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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END%%
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%%ANKI
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Basic
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The additive property applies to sets exhibiting what two properties?
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Back: Finiteness and disjointedness.
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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: 1708217738473-->
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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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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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END%%
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%%ANKI
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Basic
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How do we denote $A$ and $B$ are disjoint using standard set notation?
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Back: $A \cap B = \varnothing$
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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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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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## 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). |