notebook/notes/combinatorics/multiplicative-principle.md

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title TARGET DECK FILE TAGS tags
Combinatorics Obsidian::STEM combinatorics set
combinatorics
set

Overview

The multiplicative principle states that two finite sets A and B satisfy |A \times B| = |A| \cdot |B|

This can be generalized to any number of finite sets in the obvious way.

%%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.

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%%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.

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%%ANKI Cloze {union} is to the additive property whereas {struct} is to the multiplicative property. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: c

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%%ANKI Basic Which C construct corresponds to the multiplicative property? Back: struct Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: c

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%%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.

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%%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.

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%%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.

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%%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.

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%%ANKI Basic How many functions exist between \{1, 2, 3, 4, 5\} and \{a, b, c, d\}? Back: 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4^5 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How many functions exist between finite sets A and B? Back: |B|^{|A|} Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What combinatorial concept explains the number of functions between two finite sets? Back: The multiplicative principle. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How is the "count of three letter license plates" reimagined as a count of functions? Back: As the number of functions from \{1, 2, 3\} to \{A, B, \ldots, Z\}. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How is the "maximum unsigned w-bit number" reimagined as a count of functions? Back: As one less than the number of functions from \{1, 2, \ldots, w\} to \{0, 1\}. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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References