193 lines
5.8 KiB
Markdown
193 lines
5.8 KiB
Markdown
---
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title: Algebra of Sets
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::set set
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tags:
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- algebra
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- set
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---
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## Overview
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The study of the operations of union ($\cup$), intersection ($\cap$), and set difference ($-$), together with the inclusion relation ($\subseteq$), goes by the **algebra of sets**.
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%%ANKI
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Basic
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What three operators make up the algebra of sets?
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Back: $\cup$, $\cap$, and $-$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060602-->
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END%%
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%%ANKI
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Basic
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What *relation* is relevant in the algebra of sets?
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Back: $\subseteq$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060605-->
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END%%
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## Laws
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The algebra of sets obey laws reminiscent (but not exactly) of the algebra of real numbers.
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%%ANKI
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Cloze
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{$\cup$} is to algebra of sets whereas {$+$} is to algebra of real numbers.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060607-->
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END%%
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%%ANKI
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Cloze
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{$\cap$} is to algebra of sets whereas {$\cdot$} is to algebra of real numbers.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060609-->
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END%%
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%%ANKI
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Cloze
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{$-$} is to algebra of sets whereas {$-$} is to algebra of real numbers.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060611-->
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END%%
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%%ANKI
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Cloze
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{$\subseteq$} is to algebra of sets whereas {$\leq$} is to algebra of real numbers.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060614-->
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END%%
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### Commutative Laws
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For any sets $A$ and $B$, $$\begin{align*} A \cup B & = B \cup A \\ A \cap B & = B \cap A \end{align*}$$
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%%ANKI
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Basic
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The commutative laws of the algebra of sets apply to what operators?
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Back: $\cup$ and $\cap$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060616-->
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END%%
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%%ANKI
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Basic
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What does the union commutative law state?
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Back: For any sets $A$ and $B$, $A \cup B = B \cup A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060618-->
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END%%
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%%ANKI
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Basic
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What does the intersection commutative law state?
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Back: For any sets $A$ and $B$, $A \cap B = B \cap A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060620-->
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END%%
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### Associative Laws
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For any sets $A$ and $B$, $$\begin{align*} A \cup (B \cup C) & = (A \cup B) \cup C \\ A \cap (B \cap C) & = (A \cap B) \cap C \end{align*}$$
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%%ANKI
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Basic
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The associative laws of the algebra of sets apply to what operators?
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Back: $\cup$ and $\cap$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060622-->
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END%%
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%%ANKI
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Basic
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What does the union associative law state?
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Back: For any sets $A$, $B$, and $C$, $A \cup (B \cup C) = (A \cup B) \cup C$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060624-->
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END%%
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%%ANKI
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Basic
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What does the intersection associative law state?
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Back: For any sets $A$, $B$, and $C$, $A \cap (B \cap C) = (A \cap B) \cap C$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716396060625-->
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END%%
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### Distributive Laws
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For any sets $A$, $B$, and $C$, $$\begin{align*} A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) & = (A \cup B) \cap (A \cup C) \end{align*}$$
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%%ANKI
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Basic
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The distributive laws of the algebra of sets apply to what operators?
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Back: $\cup$ and $\cap$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270441-->
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END%%
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%%ANKI
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Cloze
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The distributive law states {$A \cap (B \cup C)$} $=$ {$(A \cap B) \cup (A \cap C)$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270447-->
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END%%
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%%ANKI
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Cloze
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The distributive law states {$A \cup (B \cap C)$} $=$ {$(A \cup B) \cap (A \cup C)$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270452-->
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END%%
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### De Morgan's Laws
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For any sets $A$, $B$, and $C$, $$\begin{align*} C - (A \cup B) & = (C - A) \cap (C - B) \\ C - (A \cap B) & = (C - A) \cup (C - B) \end{align*}$$
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%%ANKI
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Basic
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The De Morgan's laws of the algebra of sets apply to what operators?
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Back: $\cup$, $\cap$, and $-$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270457-->
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END%%
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%%ANKI
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Cloze
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De Morgan's law states that {$C - (A \cup B)$} $=$ {$(C - A) \cap (C - B)$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270461-->
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END%%
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%%ANKI
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Cloze
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De Morgan's law states that {$C - (A \cap B)$} $=$ {$(C - A) \cup (C - B)$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270466-->
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END%%
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%%ANKI
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Cloze
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For their respective De Morgan's laws, {$-$} is to the algebra of sets whereas {$\neg$} is to boolean algebra.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270473-->
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END%%
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%%ANKI
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Cloze
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For their respective De Morgan's laws, {$\cup$} is to the algebra of sets whereas {$\lor$} is to boolean algebra.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270480-->
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END%%
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%%ANKI
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Cloze
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For their respective De Morgan's laws, {$\cap$} is to the algebra of sets whereas {$\land$} is to boolean algebra.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803270485-->
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END%%
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## Bibliography
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* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |