notebook/notes/set/relations.md

title	TARGET DECK	FILE TAGS	tags
Relations	Obsidian::STEM	set::relation	relation set

Overview

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.

%%ANKI Basic How is an ordered pair of x and y denoted? Back: \langle x, y \rangle Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic What property must any satisfactory definition of \langle x, y \rangle satisfy? Back: x and y, along with their order, are uniquely determined. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic Which of ordered pairs or sets is more general? Back: Sets.

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%%ANKI Basic What biconditional is used to prove the well-definedness of \langle x, y \rangle? Back: (\langle x, y \rangle = \langle u, v \rangle) \Leftrightarrow (x = u \land y = v) Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Cloze {\{1, 2\}} is a set whereas {\langle 1, 2 \rangle} is an ordered pair. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is \langle x, y \rangle most commonly defined? Back: As \{\{x\}, \{x, y\}\}. Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.

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%%ANKI Basic Who is usually attributed the most commonly used definition of an ordered pair? Back: Kazimierz Kuratowski. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is \{\{x\}, \{x, y\}\} alternatively denoted? Back: \langle x, y \rangle Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Cloze Well-definedness of ordered pairs: {\langle u, v \rangle = \langle x, y \rangle} if and only if {u = x \land v = y}. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic What term is used to refer to x in \langle x, y \rangle? Back: The first coordinate. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Cloze y is the {second} coordinate of \langle x, y \rangle. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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A relation R is a set of ordered pairs. The domain of R (\mathop{\text{dom}}{R}), the range of R (\mathop{\text{ran}}{R}), and the field of R (\mathop{\text{fld}}{R}) is defined as:

• x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R
• x \in \mathop{\text{ran}}{R} \Leftrightarrow \exists t, \langle t, x \rangle \in R
• \mathop{\text{fld}}{R} = \mathop{\text{dom}}{R} \cup \mathop{\text{ran}}{R}

%%ANKI Basic What is a relation? Back: A set of ordered pairs. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic Are relations or sets the more general concept? Back: Sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the ordering relation < on \{2, 3, 5\} defined? Back: As set \{\langle 2, 3\rangle, \langle 2, 5 \rangle, \langle 3, 5 \rangle\}. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the ordering relation < on \{2, 3, 5\} visualized? Back: ! Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic A relation is a set of ordered pairs with what additional restriction? Back: N/A. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Cloze For relation R, {xRy} is alternative notation for {\langle x, y \rangle \in R}. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is ordering relation < on set \mathbb{R} defined using set-builder notation? Back: As \{\langle x, y\rangle \in \mathbb{R} \times \mathbb{R} \mid x \text{ is less than } y\}. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is x < y rewritten to emphasize that < is a relation? Back: \langle x, y \rangle \in \;< Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the identity relation on \omega defined using set-builder notation? Back: \{\langle n, n \rangle \mid n \in \omega\} Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the domain of relation R denoted? Back: \mathop{\text{dom}}{R} Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the domain of relation R defined? Back: x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the range of relation R denoted? Back: \mathop{\text{ran}}{R} Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the range of relation R defined? Back: x \in \mathop{\text{ran}}{R} \Leftrightarrow \exists t, \langle t, x \rangle \in R Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the field of relation R denoted? Back: \mathop{\text{fld}}{R} Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the field of relation R defined? Back: \mathop{\text{fld}}{R} = \mathop{\text{dom}}{R} \cup \mathop{\text{ran}}{R} Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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Bibliography

• “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
• Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).