1220 lines
41 KiB
Markdown
1220 lines
41 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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## Symmetric Difference
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Define the **symmetric difference** of sets $A$ and $B$ as $$A \mathop{\triangle} B = (A - B) \cup (B - A)$$
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%%ANKI
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Basic
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What two operators are used in the definition of the symmetric difference?
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Back: $\cup$ and $-$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717554445662-->
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END%%
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%%ANKI
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Basic
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How is the symmetric difference of sets $A$ and $B$ denoted?
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Back: $A \mathop{\triangle} B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717554445665-->
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END%%
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%%ANKI
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Basic
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How is $A \mathop{\triangle} B$ defined?
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Back: As $(A - B) \cup (B - A)$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717554445670-->
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END%%
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## Cartesian Product
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Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$
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%%ANKI
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Basic
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How is the Cartesian product of $A$ and $B$ denoted?
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Back: $A \times B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717679397781-->
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END%%
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%%ANKI
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Basic
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Using ordered pairs, how is $A \times B$ defined?
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Back: $\{\langle x, y \rangle \mid x \in A \land y \in B\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717679397797-->
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END%%
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%%ANKI
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Basic
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Who is attributed the representation of points in a plane?
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Back: René Descartes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717679397825-->
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END%%
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%%ANKI
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Basic
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Why is the Cartesian product named the way it is?
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Back: It is named after René Descartes.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1717679397836-->
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END%%
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%%ANKI
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Basic
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Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
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Back: $\mathscr{P}\mathscr{P}A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717679397848-->
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END%%
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%%ANKI
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Cloze
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{$x \in A$} iff {$\{x\} \subseteq A$} iff {$\{x\} \in \mathscr{P}A$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717679397860-->
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END%%
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We can also form (something like) the Cartesian product of infinitely many sets, provided that the sets are suitably indexed. Let $I$ be an index set and $H$ a function whose domain includes $I$. Define $$\bigtimes_{i \in I} H(i) = \{f \mid f \text{ is a function with domain } I \text{ and } \forall i \in I, f(i) \in H(i)\}$$
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%%ANKI
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Basic
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What kind of mathematical object is $I$ in $\bigtimes_{i \in I} H(i)$?
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Back: A set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209655-->
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END%%
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%%ANKI
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Basic
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What kind of mathematical object is $H$ in $\bigtimes_{i \in I} H(i)$?
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Back: A function.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209661-->
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END%%
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%%ANKI
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Basic
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What is the domain of $H$ in $\bigtimes_{i \in I} H(i)$?
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Back: Some superset of $I$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209666-->
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END%%
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%%ANKI
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Basic
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What is the range of $H$ in $\bigtimes_{i \in I} H(i)$?
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Back: Some superset of $\{H(i) \mid i \in I\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209672-->
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END%%
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%%ANKI
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Basic
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Let $I$ be an index set and $H$ a function $I \subseteq \mathop{\text{dom}}H$. How is $\bigtimes_{i \in I} H(i)$ defined?
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Back: $\bigtimes_{i \in I} H(i) = \{ f \mid f \text{ is a function with domain } I \text { and } \forall i \in I, f(i) \in H(i) \}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209677-->
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END%%
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%%ANKI
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Basic
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What kind of mathematical object is $h \in \bigtimes_{i \in I} H(i)$?
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Back: A function.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209682-->
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END%%
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%%ANKI
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Basic
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Let $f \in \bigtimes_{i \in I} H(i)$. What is the domain of $f$?
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Back: $I$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209686-->
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END%%
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%%ANKI
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Basic
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Let $f \in \bigtimes_{i \in I} H(i)$. What is the codomain of $f$?
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Back: $\bigcup_{i \in I} H(i)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209690-->
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END%%
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%%ANKI
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Basic
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Given arbitrary sets $A$ and $B$, what index set $I$ and function $H$ satisfies $A \times B = \bigtimes_{i \in I} H(i)$?
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209694-->
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END%%
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%%ANKI
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Basic
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*Why* can't $A \times B$ be rewritten with $\bigtimes_{i \in I} H(i)$ assuming suitable $I$ and $H$?
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Back: The former is a set of ordered pairs. The latter is a set of functions.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209698-->
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END%%
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%%ANKI
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Basic
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Assume AoC and $H(j) = \varnothing$ for some $j \in I$. What does $\bigtimes_{i \in I} H(i)$ evaluate to?
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Back: $\varnothing$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209702-->
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END%%
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%%ANKI
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Basic
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When does $\bigtimes_{i \in I} H(i) = \varnothing$?
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Back: When there exists some $i \in I$ such that $H(i) = \varnothing$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209705-->
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END%%
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%%ANKI
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Basic
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Assume AoC and $H(j) \neq \varnothing$ for all $j \in I$. What does $\bigtimes_{i \in I} H(i)$ evaluate to?
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Back: A non-empty set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209709-->
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END%%
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%%ANKI
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Basic
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The following is likely a diagram of what?
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![[infinite-cartesian-product.png]]
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Back: A member of $\bigtimes_{i \in \omega} H(i)$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209713-->
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END%%
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%%ANKI
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Basic
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Suppose $H(i) \neq \varnothing$ for all $i \in I$. When is $\bigtimes_{i \in I} H(i) \neq \varnothing$?
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Back: When AoC is included in our formal system.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720964209716-->
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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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%%ANKI
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Basic
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Is the Cartesian product commutative?
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Back: No.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881694-->
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END%%
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%%ANKI
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Basic
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*Why* isn't the Cartesian product commutative?
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Back: Because the Cartesian product comprises of *ordered* pairs.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881698-->
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END%%
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%%ANKI
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Basic
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Suppose $A \neq \varnothing$ and $B \neq \varnothing$. When does $A \times B = B \times A$?
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Back: When $A = B$.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881702-->
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END%%
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%%ANKI
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Basic
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Suppose $A \neq \varnothing$ and $A \neq B$. When does $A \times B = B \times A$?
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Back: When $B = \varnothing$.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881705-->
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END%%
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%%ANKI
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Basic
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Under what two conditions is $A \times B = B \times A$?
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Back: $A = B$ or either set is the empty set.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881709-->
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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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%%ANKI
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Basic
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Is the Cartesian product associative?
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Back: No.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881712-->
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END%%
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%%ANKI
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Basic
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*Why* isn't the Cartesian product associative?
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Back: The association of parentheses defines the nesting of the ordered pairs.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881715-->
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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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%%ANKI
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Basic
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What concept in set theory relates the algebra of sets to boolean algebra?
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Back: Membership.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717372322271-->
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END%%
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%%ANKI
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Basic
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What two equalities relates $A \cup B$ with $a \lor b$?
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Back: $a = (x \in A)$ and $b = (x \in B)$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717372322264-->
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END%%
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%%ANKI
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Basic
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What two equalities relates $A \cap B$ with $a \land b$?
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Back: $a = (x \in A)$ and $b = (x \in B)$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717372322275-->
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END%%
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More generally, for any sets $A$ and $\mathscr{B}$, $$\begin{align*} A \cup \bigcap \mathscr{B} & = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}, \text{ for } \mathscr{B} \neq \varnothing \\ A \cap \bigcup \mathscr{B} & = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\} \end{align*}$$
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%%ANKI
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Basic
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What is the generalization of identity $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$?
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Back: $A \cap \bigcup \mathscr{B} = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717366846568-->
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END%%
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%%ANKI
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Basic
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What is the generalization of identity $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$?
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Back: $A \cup \bigcap \mathscr{B} = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ for $\mathscr{B} \neq \varnothing$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717366846580-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
Assuming $\mathscr{B} \neq \varnothing$, the distributive law states {$A \cup \bigcap \mathscr{B}$} $=$ {$\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717366846573-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
The distributive law states {$A \cap \bigcup \mathscr{B}$} $=$ {$\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717366846594-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is set $\{A \cup X \mid X \in \mathscr{B}\}$ pronounced?
|
|
Back: The set of all $A \cup X$ such that $X \in \mathscr{B}$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717367767303-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the specialization of identity $A \cap \bigcup \mathscr{B} = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$?
|
|
Back: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717367767308-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the specialization of identity $A \cup \bigcap \mathscr{B} = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$?
|
|
Back: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717367767311-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does $\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B}$ gets larger?
|
|
Back: Larger.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717372322278-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does $\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B}$ gets smaller?
|
|
Back: Smaller.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717372322281-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does $\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B} \neq \varnothing$ gets larger?
|
|
Back: Smaller.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717372322284-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does $\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B} \neq \varnothing$ gets smaller?
|
|
Back: Larger.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717372322287-->
|
|
END%%
|
|
|
|
For any sets $A$, $B$, and $C$, $$\begin{align*} A \times (B \cap C) & = (A \times B) \cap (A \times C) \\ A \times (B \cup C) & = (A \times B) \cup (A \times C) \\ A \times (B - C) & = (A \times B) - (A \times C) \end{align*}$$
|
|
%%ANKI
|
|
Basic
|
|
Which of the algebra of sets operators does the Cartesian product distributive over?
|
|
Back: $\cap$, $\cup$, and $-$
|
|
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1718069881718-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What distributivity rule is satisfied by $\cap$ and $\times$?
|
|
Back: $A \times (B \cap C) = (A \times B) \cap (A \times C)$
|
|
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1718069881723-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
The Cartesian product satisfies distributivity: {$A \times (B \cap C)$} $=$ {$(A \times B) \cap (A \times C)$}.
|
|
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1718069881726-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What distributivity rule is satisfied by $\cup$ and $\times$?
|
|
Back: $A \times (B \cup C) = (A \times B) \cup (A \times C)$
|
|
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1718069881731-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
The Cartesian product satisfies distributivity: {$A \times (B \cup C)$} $=$ {$(A \times B) \cup (A \times C)$}.
|
|
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1718069881735-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What distributivity rule is satisfied by $-$ and $\times$?
|
|
Back: $A \times (B - C) = (A \times B) - (A \times C)$
|
|
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1718069881742-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
The Cartesian product satisfies distributivity: {$A \times (B - C)$} $=$ {$(A \times B) - (A \times C)$}.
|
|
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1718069881752-->
|
|
END%%
|
|
|
|
In addition, $$\begin{align*} A \times \bigcup \mathscr{B} & = \bigcup\, \{A \times X \mid X \in \mathscr{B}\} \\ A \times \bigcap \mathscr{B} & = \bigcap\, \{A \times X \mid X \in \mathscr{B}\} \end{align*}$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the generalization of identity $A \times (B \cup C) = (A \times B) \cup (A \times C)$?
|
|
Back: $A \times \bigcup \mathscr{B} = \bigcup\, \{A \times X \mid X \in \mathscr{B}\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718069881759-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the specialization of identity $A \times \bigcap \mathscr{B} = \bigcap\, \{A \times X \mid X \in \mathscr{B}\}$?
|
|
Back: $A \times (B \cap C) = (A \times B) \cap (A \times C)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718069881766-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the generalization of identity $A \times (B \cap C) = (A \times B) \cap (A \times C)$?
|
|
Back: $A \times \bigcap \mathscr{B} = \bigcap\, \{A \times X \mid X \in \mathscr{B}\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718069881773-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the specialization of identity $A \times \bigcup \mathscr{B} = \bigcup\, \{A \times X \mid X \in \mathscr{B}\}$?
|
|
Back: $A \times (B \cup C) = (A \times B) \cup (A \times C)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718069881779-->
|
|
END%%
|
|
|
|
### De Morgan's Laws
|
|
|
|
For any sets $A$, $B$, and $C$, $$\begin{align*} C - (A \cup B) & = (C - A) \cap (C - B) \\ C - (A \cap B) & = (C - A) \cup (C - B) \end{align*}$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
The De Morgan's laws of the algebra of sets apply to what operators?
|
|
Back: $\cup$, $\cap$, and $-$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1716803270457-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
De Morgan's law states that {$C - (A \cup B)$} $=$ {$(C - A) \cap (C - B)$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1716803270461-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
De Morgan's law states that {$C - (A \cap B)$} $=$ {$(C - A) \cup (C - B)$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1716803270466-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For their respective De Morgan's laws, {$-$} is to the algebra of sets whereas {$\neg$} is to boolean algebra.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1716803270473-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For their respective De Morgan's laws, {$\cup$} is to the algebra of sets whereas {$\lor$} is to boolean algebra.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1716803270480-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For their respective De Morgan's laws, {$\cap$} is to the algebra of sets whereas {$\land$} is to boolean algebra.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1716803270485-->
|
|
END%%
|
|
|
|
More generally, for any sets $C$ and $\mathscr{A} \neq \varnothing$, $$\begin{align*} C - \bigcup \mathscr{A} & = \bigcap\, \{C - X \mid X \in \mathscr{A}\} \\ C - \bigcap \mathscr{A} & = \bigcup\, \{C - X \mid X \in \mathscr{A}\} \end{align*}$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the generalization of identity $C - (A \cup B) = (C - A) \cap (C - B)$?
|
|
Back: $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717367767316-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the generalization of identity $C - (A \cap B) = (C - A) \cup (C - B)$?
|
|
Back: $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717367767323-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For $\mathscr{A} \neq \varnothing$, De Morgan's law states that {$C - \bigcap \mathscr{A}$} $=$ {$\bigcup\, \{C - X \mid X \in \mathscr{A}\}$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717367767320-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the specialization of identity $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$?
|
|
Back: $C - (A \cup B) = (C - A) \cap (C - B)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717373048517-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the specialization of identity $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$?
|
|
Back: $C - (A \cap B) = (C - A) \cup (C - B)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717373048522-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which law of the algebra of sets is represented by e.g. $C - (A \cup B) = (C - A) \cap (C - B)$?
|
|
Back: De Morgan's Law.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717373048525-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For $\mathscr{A} \neq \varnothing$, De Morgan's law states that {$C - \bigcup \mathscr{A}$} $=$ {$\bigcap\, \{C - X \mid X \in \mathscr{A}\}$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717367767328-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why does identity $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$ fail when $\mathscr{A} = \varnothing$?
|
|
Back: The RHS evaluates to class $\bigcap \varnothing$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717368301050-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why does identity $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$ fail when $\mathscr{A} = \varnothing$?
|
|
Back: $\bigcap \mathscr{A}$ is undefined.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717368301055-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does $\bigcap\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets larger?
|
|
Back: Smaller.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717372322295-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does $\bigcap\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets smaller?
|
|
Back: Larger.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717372322299-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Does $\bigcup\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets larger?
|
|
Back: Larger.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717372322304-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Does $\bigcup\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets smaller?
|
|
Back: Smaller.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
END%%
|
|
|
|
### Monotonicity
|
|
|
|
Let $A$, $B$, and $C$ be arbitrary sets. Then
|
|
|
|
* $A \subseteq B \Rightarrow A \cup C \subseteq B \cup C$,
|
|
* $A \subseteq B \Rightarrow A \cap C \subseteq B \cap C$,
|
|
* $A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of propositional logical statement are the monotonicity properties of $\subseteq$?
|
|
Back: An implication.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536967-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the shared antecedent of the monotonicity properties of $\subseteq$?
|
|
Back: $A \subseteq B$ for some sets $A$ and $B$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536973-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given sets $A$, $B$, and $C$, state the monotonicity property of $\subseteq$ related to the $\cup$ operator.
|
|
Back: $A \subseteq B \Rightarrow A \cup C \subseteq B \cup C$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536976-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given sets $A$, $B$, and $C$, state the monotonicity property of $\subseteq$ related to the $\cap$ operator.
|
|
Back: $A \subseteq B \Rightarrow A \cap C \subseteq B \cap C$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536979-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given sets $A$ and $B$, state the monotonicity property of $\subseteq$ related to the $\bigcup$ operator.
|
|
Back: $A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536982-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why are the monotonicity properties of $\subseteq$ named the way they are?
|
|
Back: The ordering of operands in the antecedent are preserved in the consequent.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536985-->
|
|
END%%
|
|
|
|
In addition,
|
|
|
|
* $A \subseteq B \Rightarrow A \times C \subseteq B \times C$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What monotonicity property does the Cartesian product satisfy?
|
|
Back: $A \subseteq B \Rightarrow A \times C \subseteq B \times C$
|
|
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1718069881786-->
|
|
END%%
|
|
|
|
### Antimonotonicity
|
|
|
|
Let $A$, $B$, and $C$ be arbitrary sets. Then
|
|
|
|
* $A \subseteq B \Rightarrow C - B \subseteq C - A$,
|
|
* $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of propositional logical statement are the antimonotonicity properties of $\subseteq$?
|
|
Back: An implication.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536988-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the shared antecedent of the antimonotonicity properties of $\subseteq$?
|
|
Back: N/A.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536991-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
{1:Monotonicity} of $\subseteq$ is to {2:$\bigcup$} whereas {2:antimonotonicity} of $\subseteq$ is to {1:$\bigcap$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536994-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why are the antimonotonicity properties of $\subseteq$ named the way they are?
|
|
Back: The ordering of operands in the antecedent are reversed in the consequent.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073536998-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given sets $A$ and $B$, state the antimonotonicity property of $\subseteq$ related to the $\bigcap$ operator.
|
|
Back: $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073537001-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given sets $A$, $B$, and $C$, state the antimonotonicity property of $\subseteq$ related to the $-$ operator.
|
|
Back: $A \subseteq B \Rightarrow C - B \subseteq C - A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073537004-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why do we need the empty set check in $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$?
|
|
Back: $\bigcap A$ is not a set.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1717073537007-->
|
|
END%%
|
|
|
|
### Cancellation Laws
|
|
|
|
Let $A$, $B$, and $C$ be sets. If $A \neq \varnothing$,
|
|
|
|
* $(A \times B = A \times C) \Rightarrow B = C$
|
|
* $(B \times A = C \times A) \Rightarrow B = C$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the left cancellation law of the Cartesian product?
|
|
Back: If $A \neq \varnothing$ then $(A \times B = A \times C) \Rightarrow B = C$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718107987907-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
$(A \times B = A \times C) \Rightarrow B = C$ is always true if what condition is satisfied?
|
|
Back: $A \neq \varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718107987918-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the right cancellation law of the Cartesian product?
|
|
Back: If $A \neq \varnothing$ then $(B \times A = C \times A) \Rightarrow B = C$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718107987928-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
$(B \times A = C \times A) \Rightarrow B = C$ is always true if what condition is satisfied?
|
|
Back: $A \neq \varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718107987936-->
|
|
END%%
|
|
|
|
## Index Sets
|
|
|
|
Let $I$ be a set, called the **index set**. Let $F$ be a [[functions|function]] whose domain includes $I$. Then we define $$\bigcup_{i \in I} F(i) = \bigcup\,\{F(i) \mid i \in I\}$$
|
|
and, if $I \neq \varnothing$, $$\bigcap_{i \in I} F(i) = \bigcap\, \{F(i) \mid i \in I\}$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What name does $I$ go by in expression $\bigcup_{i \in I} F(i)$?
|
|
Back: The "index set".
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492681-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $\bigcup_{i \in I} F(i)$ alternatively denoted?
|
|
Back: $\bigcup_{i \in I} F_i$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492687-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object is $I$ in expression $\bigcup_{i \in I} F(i)$?
|
|
Back: A set.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492690-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object is $F$ in expression $\bigcup_{i \in I} F(i)$?
|
|
Back: A function.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492693-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $\bigcup_{i \in I} F_i$ alternatively denoted?
|
|
Back: $\bigcup_{i \in I} F(i)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782592276-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object is $F$ in expression $\bigcup_{i \in I} F_i$?
|
|
Back: A function.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782592281-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the domain of $F$ assumed to be in expression $\bigcup_{i \in I} F(i)$?
|
|
Back: Some superset of $I$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492696-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What condition must $I$ satisfy in expression $\bigcup_{i \in I} F(i)$?
|
|
Back: N/A.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492699-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $I = \{0, 1, 2\}$. What does $\bigcup_{i \in I} F(i)$ evaluate to?
|
|
Back: $F(0) \cup F(1) \cup F(2)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492702-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $I = \varnothing$. What does $\bigcup_{i \in I} F(i)$ evaluate to?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492705-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object is $F$ in expression $\bigcap_{i \in I} F(i)$?
|
|
Back: A function.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492709-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $\bigcap_{i \in I} F(i)$ often alternatively denoted?
|
|
Back: $\bigcap_{i \in I} F_i$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492712-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the domain of $F$ assumed to be in expression $\bigcap_{i \in I} F(i)$?
|
|
Back: Some superset of $I$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492716-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What condition must $I$ satisfy in expression $\bigcap_{i \in I} F(i)$?
|
|
Back: $I \neq \varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492720-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $I = \{0, 1, 2\}$. What does $\bigcap_{i \in I} F(i)$ evaluate to?
|
|
Back: $F(0) \cap F(1) \cap F(2)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492724-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $I = \varnothing$. What does $\bigcap_{i \in I} F(i)$ evaluate to?
|
|
Back: N/A. This is undefined.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782492727-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $\bigcap_{i \in I} F_i$ alternatively denoted?
|
|
Back: $\bigcap_{i \in I} F(i)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782592285-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical object is $F$ in expression $\bigcap_{i \in I} F_i$?
|
|
Back: A function.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782592288-->
|
|
END%%
|
|
|
|
## Function Sets
|
|
|
|
For sets $A$ and $B$, the collection of functions $F$ from $A$ into $B$ is: $$^AB = \{F \mid F \colon A \rightarrow B\}$$
|
|
$^AB$ is read as "$B$-pre-$A$". It is often written as $B^A$ instead.
|
|
|
|
%%ANKI
|
|
Basic
|
|
For sets $A$ and $B$, how is set $B^A$ defined?
|
|
Back: $\{F \mid F \colon A \rightarrow B\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782833225-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For sets $A$ and $B$, how is set $^AB$ defined?
|
|
Back: $\{F \mid F \colon A \rightarrow B\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782923177-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For any function $F \colon A \rightarrow B$, $F$ is a subset of what other set?
|
|
Back: $A \times B$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782833233-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For any function $F \colon A \rightarrow B$, $F$ is a member of what other set?
|
|
Back: $\mathscr{P}(A \times B)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782833236-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For sets $A$ and $B$, how is set $B^A$ pronounced?
|
|
Back: As "$B$-pre-$A$".
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782923183-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why prefer notation $B^A$ over $^AB$?
|
|
Back: The notation mirrors $|B|^{|A|}$, the number of elements in $B^A$ given both sets are finite.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607431-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For sets $A$ and $B$, how is set $^AB$ pronounced?
|
|
Back: As "$B$-pre-$A$".
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782923193-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why prefer notation $^AB$ over $B^A$?
|
|
Back: Because the sets are written left-to-right, from domain to codomain.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607434-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* is set $B^A$ denoted the way it is?
|
|
Back: If $A$ and $B$ are finite, then $B^A$ has $|B|^{|A|}$ members.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720782923188-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the domain of $^\omega\{0, 1\}$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607437-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the domain of a member of $^\omega\{0, 1\}$?
|
|
Back: $\omega$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607440-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the range of $\{0, 1\}^\omega$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607444-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the range of a member of $\{0, 1\}^\omega$?
|
|
Back: $\{0, 1\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607448-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does $\varnothing^\varnothing$ evaluate to?
|
|
Back: $\{\varnothing\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607451-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For $A \neq \varnothing$, what does $\varnothing^A$ evaluate to?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607455-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For $A \neq \varnothing$, *why* does $\varnothing^A = \varnothing$?
|
|
Back: No function can map a nonempty domain to an empty range.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607459-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For $A \neq \varnothing$, what does $^\varnothing A$ evaluate to?
|
|
Back: $\{\varnothing\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607463-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For $A \neq \varnothing$, *why* does $^\varnothing A = \{\varnothing\}$?
|
|
Back: $\varnothing$ is the only function with empty domain.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720783607468-->
|
|
END%%
|
|
|
|
## Bibliography
|
|
|
|
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |