144 lines
5.8 KiB
Markdown
144 lines
5.8 KiB
Markdown
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---
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title: Bags
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set::bag
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tags:
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- bag
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- set
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---
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## Overview
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Sets can contain the same element only once. This means $\{1, 1\} = \{1\}$. A **bag** (or **multiset**) can include a single element multiple times.
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%%ANKI
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Basic
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What is a bag?
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Back: A set in which we are allowed to include a single element multiple times.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266774-->
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END%%
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%%ANKI
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Basic
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What alternative term do "bags" go by?
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Back: Multisets.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266800-->
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END%%
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%%ANKI
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Cloze
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The {multiplicity} of an element in a bag is the {number of instances given for that element}.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266806-->
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END%%
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%%ANKI
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Basic
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How many bags exist with members $a$ and $b$?
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Back: An infinite amount.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266813-->
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END%%
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%%ANKI
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Basic
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What is the multiplicity of $a$ in bag $\{a, b\}$?
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Back: $1$
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266819-->
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END%%
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%%ANKI
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Basic
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What is the multiplicity of $a$ in bag $\{a, a, a, b, b, c\}$?
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Back: $3$
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266825-->
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END%%
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%%ANKI
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Basic
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What distinguishes a bag from a tuple?
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Back: Order is irrelevant in the former.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266830-->
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END%%
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%%ANKI
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Basic
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What is the cardinality of $a$ in bag $\{a, a, a, b, b, c\}$?
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Back: N/A. Cardinality applies to bags, not members.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266835-->
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END%%
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%%ANKI
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Basic
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What is the cardinality of bag $\{a, a, a, b, b, c\}$?
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Back: $6$
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266840-->
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END%%
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%%ANKI
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Basic
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How is the cardinality of a bag calculated?
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Back: By summing the multiplicities of each member.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266845-->
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END%%
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%%ANKI
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Basic
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When viewed as a function, a multiset has domain what?
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Back: The set comprising each member of the multiset.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266850-->
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END%%
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%%ANKI
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Basic
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When viewed as a function, a multiset has range what?
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Back: The set comprising the multiplicities of the multiset's members.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266856-->
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END%%
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%%ANKI
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Basic
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When viewed as a function, a multiset has codomain what?
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Back: The positive integers.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266861-->
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END%%
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%%ANKI
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Basic
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Do all multisets correspond to sets?
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Back: No.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266867-->
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END%%
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%%ANKI
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Basic
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Do all sets correspond to multisets?
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Back: Yes.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266873-->
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END%%
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%%ANKI
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Basic
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Under what condition is a bag also a set?
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Back: When the multiplicity of each element in the bag is $1$.
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Reference: “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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<!--ID: 1720360266878-->
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END%%
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## Bibliography
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* “Multiset,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725](https://en.wikipedia.org/w/index.php?title=Multiset&oldid=1217165725).
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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).
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