149 lines
4.3 KiB
Markdown
149 lines
4.3 KiB
Markdown
---
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title: Relations
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set::relation
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tags:
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- relation
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- set
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---
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## Overview
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An ordered pair of $x$ and $y$, denoted $\langle x, y \rangle$, is defined as: $\langle x, y \rangle = \{\{x\}, \{x, y\}\}$. We define the **first coordinate** of $\langle x, y \rangle$ to be $x$ and the **second coordinate** to be $y$.
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%%ANKI
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Basic
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How is an ordered pair of $x$ and $y$ denoted?
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Back: $\langle x, y \rangle$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753102-->
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END%%
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%%ANKI
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Basic
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What property must any satisfactory definition of $\langle x, y \rangle$ satisfy?
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Back: $x$ and $y$, along with their order, are uniquely determined.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717679524930-->
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END%%
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%%ANKI
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Basic
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Which of ordered pairs or sets is more general?
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Back: Sets.
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<!--ID: 1717678753108-->
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END%%
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%%ANKI
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Basic
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What biconditional is used to prove the well-definedness of $\langle x, y \rangle$?
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Back: $(\langle x, y \rangle = \langle u, v \rangle) \Leftrightarrow (x = u \land y = v)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753111-->
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END%%
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%%ANKI
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Cloze
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{$\{1, 2\}$} is a set whereas {$\langle 1, 2 \rangle$} is an ordered pair.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753116-->
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END%%
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%%ANKI
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Basic
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How is $\langle x, y \rangle$ usually defined?
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Back: As $\{\{x\}, \{x, y\}\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753120-->
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END%%
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%%ANKI
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Basic
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Who is usually attributed the most commonly used definition of an ordered pair?
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Back: Kazimierz Kuratowski.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753124-->
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END%%
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%%ANKI
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Basic
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How is $\{\{x\}, \{x, y\}\}$ alternatively denoted?
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Back: $\langle x, y \rangle$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753129-->
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END%%
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%%ANKI
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Cloze
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Well-definedness of ordered pairs: {$\langle u, v \rangle = \langle x, y \rangle$} if and only if {$u = x \land v = y$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753134-->
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END%%
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%%ANKI
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Basic
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What term is used to refer to $x$ in $\langle x, y \rangle$?
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Back: The first coordinate.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753139-->
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END%%
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%%ANKI
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Cloze
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$y$ is the {second} coordinate of $\langle x, y \rangle$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717678753145-->
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END%%
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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: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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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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## Bibliography
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* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |