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| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Multiplicative Principle | Obsidian::STEM | 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.
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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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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: c17
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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: c17
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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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Basic
If A is finite, how is A \times B rewritten as the union of |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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Basic
If B is finite, how is A \times B rewritten as the union of |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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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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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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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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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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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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Bibliography
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.