1344 lines
48 KiB
Markdown
1344 lines
48 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 $\langle x, y \rangle$ most commonly defined?
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Back: As $\{\{x\}, \{x, y\}\}$.
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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: 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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%%ANKI
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Basic
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Is $\varnothing$ a relation?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1719681913524-->
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END%%
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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:
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* $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$
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* $x \in \mathop{\text{ran}}{R} \Leftrightarrow \exists t, \langle t, x \rangle \in R$
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* $\mathop{\text{fld}}{R} = \mathop{\text{dom}}{R} \cup \mathop{\text{ran}}{R}$
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%%ANKI
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Basic
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What is a relation?
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Back: A set of ordered pairs.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987764-->
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END%%
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%%ANKI
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Basic
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Are relations or sets the more general concept?
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Back: Sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987776-->
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END%%
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%%ANKI
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Basic
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How is the ordering relation $<$ on $\{2, 3, 5\}$ defined?
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Back: As set $\{\langle 2, 3\rangle, \langle 2, 5 \rangle, \langle 3, 5 \rangle\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987783-->
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END%%
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%%ANKI
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Basic
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How is the ordering relation $<$ on $\{2, 3, 5\}$ visualized?
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Back:
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![[relation-ordering-example.png]]
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987794-->
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END%%
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%%ANKI
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Basic
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A relation is a set of ordered pairs with what additional restriction?
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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: 1718107987803-->
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END%%
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%%ANKI
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Cloze
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For relation $R$, {$xRy$} is alternative notation for {$\langle x, y \rangle \in R$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987813-->
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END%%
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%%ANKI
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Basic
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How is ordering relation $<$ on set $\mathbb{R}$ defined using set-builder notation?
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Back: As $\{\langle x, y\rangle \in \mathbb{R} \times \mathbb{R} \mid x \text{ is less than } y\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987822-->
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END%%
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%%ANKI
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Basic
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How is $x < y$ rewritten to emphasize that $<$ is a relation?
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Back: $\langle x, y \rangle \in \;<$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987831-->
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END%%
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%%ANKI
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Basic
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How is the identity relation on $\omega$ defined using set-builder notation?
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Back: $\{\langle n, n \rangle \mid n \in \omega\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987840-->
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END%%
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%%ANKI
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Basic
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How is the domain of relation $R$ denoted?
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Back: $\mathop{\text{dom}}{R}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987850-->
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END%%
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%%ANKI
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Basic
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How is the domain of relation $R$ defined?
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Back: $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987862-->
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END%%
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%%ANKI
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Basic
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What is the most general mathematical object the $\mathop{\text{dom}}$ operation can be applied to?
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Back: Sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718546439334-->
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END%%
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%%ANKI
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Basic
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Let $A$ be a set containing no ordered pairs. What is $\mathop{\text{dom}} A$?
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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: 1718327739893-->
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END%%
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%%ANKI
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Basic
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Let $A = \{\{\{x\}, \{x, y\}\}, \{z\}\}$. What is $\mathop{\text{dom}} A$?
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Back: $\{x\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739898-->
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END%%
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%%ANKI
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Cloze
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For any set $\mathscr{A}$, $\mathop{\text{dom}}\bigcup\mathscr{A}$ {$=$} $\bigcup\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739901-->
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END%%
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%%ANKI
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Basic
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The following is analagous to what logical expression of commuting quantifiers?$$\mathop{\text{dom}}\bigcup\mathscr{A} = \bigcup\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
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Back: $\exists x, \exists y, P(x, y) \Leftrightarrow \exists y, \exists x, P(x, y)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739907-->
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END%%
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%%ANKI
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Cloze
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For any set $\mathscr{A}$, $\mathop{\text{dom}}\bigcap\mathscr{A}$ {$\subseteq$} $\bigcap\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739910-->
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END%%
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%%ANKI
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Basic
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The following is analagous to what logical expression of commuting quantifiers? $$\mathop{\text{dom}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
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Back: $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739914-->
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END%%
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%%ANKI
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Cloze
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For any set $\mathscr{A}$, $\mathop{\text{ran}}\bigcup\mathscr{A}$ {$=$} $\bigcup\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739918-->
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END%%
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%%ANKI
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Basic
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The following is analagous to what predicate logical expression of commuting quantifiers? $$\mathop{\text{ran}}\bigcup\mathscr{A} = \bigcup\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$$
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Back: $\exists x, \exists y, P(x, y) \Leftrightarrow \exists y, \exists x, P(x, y)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739922-->
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END%%
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%%ANKI
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Cloze
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For any set $\mathscr{A}$, $\mathop{\text{ran}}\bigcap\mathscr{A}$ {$\subseteq$} $\bigcap\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739926-->
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END%%
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%%ANKI
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Basic
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The following is analagous to what logical expression of commuting quantifiers? $$\mathop{\text{ran}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$$
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Back: $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739931-->
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END%%
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%%ANKI
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Basic
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How is the range of relation $R$ denoted?
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Back: $\mathop{\text{ran}}{R}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987872-->
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END%%
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%%ANKI
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Basic
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How is the range of relation $R$ defined?
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Back: $x \in \mathop{\text{ran}}{R} \Leftrightarrow \exists t, \langle t, x \rangle \in R$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987880-->
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END%%
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%%ANKI
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Basic
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What is the most general mathematical object the $\mathop{\text{ran}}$ operation can be applied to?
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Back: Sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718546439338-->
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END%%
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%%ANKI
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Basic
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Let $A$ be a set containing no ordered pairs. What is $\mathop{\text{ran}} A$?
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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: 1718327739936-->
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END%%
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%%ANKI
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Basic
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Let $A = \{\{\{x\}, \{x, y\}\}, \{z\}\}$. What is $\mathop{\text{ran}} A$?
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Back: $\{y\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739940-->
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END%%
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%%ANKI
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Basic
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How is the field of relation $R$ denoted?
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Back: $\mathop{\text{fld}}{R}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987887-->
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END%%
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%%ANKI
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Basic
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How is the field of relation $R$ defined?
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Back: $\mathop{\text{fld}}{R} = \mathop{\text{dom}}{R} \cup \mathop{\text{ran}}{R}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718107987897-->
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END%%
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%%ANKI
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Basic
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Let $A = \{\{\{x\}, \{x, y\}\}, \{z\}\}$. What is $\mathop{\text{fld}} A$?
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Back: $\{x, y\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739945-->
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END%%
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%%ANKI
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Basic
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If $\langle x, y \rangle \in A$, what sets are in $\bigcup A$?
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Back: $\{x\}$ and $\{x, y\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739950-->
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END%%
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%%ANKI
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Basic
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If $\langle x, y \rangle \in A$, what sets are in $\bigcup \bigcup A$?
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Back: $x$ and $y$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739955-->
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END%%
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%%ANKI
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Basic
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What is the most general mathematical object the $\mathop{\text{fld}}$ operation can be applied to?
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Back: Sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718546439341-->
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END%%
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%%ANKI
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Basic
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$\mathop{\text{fld}} R = \bigcup \bigcup R$ is necessary for what condition?
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Back: $R$ is a relation.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718327739961-->
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END%%
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A set $A$ is **single-valued** iff for each $x$ in $\mathop{\text{dom}}A$, there is only one $y$ such that $xAy$. A set $A$ is **single-rooted** iff for each $y \in \mathop{\text{ran}}A$, there is only one $x$ such that $xAy$.
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%%ANKI
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Basic
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What does it mean for a set $A$ to be "single-valued"?
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Back: For each $x \in \mathop{\text{dom}}A$, there exists a unique $y$ such that $xAy$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718427443355-->
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END%%
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%%ANKI
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Basic
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What does it mean for a set $A$ to be "single-rooted"?
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Back: For each $y \in \mathop{\text{ran}}A$, there exists a unique $x$ such that $xAy$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718465870483-->
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END%%
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%%ANKI
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Cloze
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For any binary relation $R$, $R \subseteq$ {1:$\mathop{\text{dom} }R$} $\times$ {1:$\mathop{\text{ran} }R$} $\subseteq$ {2:$\mathop{\text{fld} }R$} $\times$ {2:$\mathop{\text{fld} }R$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720991126990-->
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END%%
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## n-ary Relations
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We define ordered triples as $\langle x, y, z \rangle = \langle \langle x, y \rangle, z \rangle$. We define ordered quadruples as $\langle x_1, x_2, x_3, x_4 \rangle = \langle \langle \langle x_1, x_2 \rangle, x_3 \rangle, x_4 \rangle$. This idea generalizes to $n$-tuples. As a special case, we define the $1$-tuple $\langle x \rangle = x$.
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An **$n$-ary relation on $A$** is a set of ordered $n$-tuples with all **components** in $A$. Keep in mind though, a unary ($1$-ary) relation on $A$ is just a subset of $A$ and may not be a relation at all.
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%%ANKI
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Basic
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Ordered triple $\langle x, y, z \rangle$ is "syntactic sugar" for what?
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Back: $\langle \langle x, y \rangle, z \rangle$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718329620058-->
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END%%
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%%ANKI
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Basic
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Ordered quadruple $\langle x_1, x_2, x_3, x_4 \rangle$ is "syntactic sugar" for what?
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Back: $\langle \langle \langle x_1, x_2 \rangle, x_3 \rangle, x_4 \rangle$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718329620086-->
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END%%
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%%ANKI
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Basic
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A $1$-tuple $\langle x \rangle$ is "syntactic sugar" for what?
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Back: $x$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718329620091-->
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END%%
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%%ANKI
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Basic
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What simpler construct are $n$-tuples constructed from?
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Back: Ordered pairs.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718329620096-->
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END%%
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%%ANKI
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Basic
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Are $n$-tuples defined in a left- or right-associative way?
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Back: Left-associative.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620101-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*What* is an $n$-tuple?
|
|
Back: A left-associative nesting of $n$ elements as ordered pairs.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620108-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*What* is an $n$-ary relation on $A$?
|
|
Back: A set of ordered $n$-tuples with all components in $A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620114-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does it mean for a relation to be on some set $A$?
|
|
Back: The components of the relation's members are members of $A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718427443424-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
A $2$-ary relation on $A$ is a subset of what Cartesian product?
|
|
Back: $A \times A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620119-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
A $1$-ary relation on $A$ is a subset of what Cartesian product?
|
|
Back: N/A.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620126-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
A $3$-ary relation on $A$ is a subset of what Cartesian product?
|
|
Back: $(A \times A) \times A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620132-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What terminological quirk exists with respect to $n$-ary relations on $A$?
|
|
Back: A $1$-ary relation on $A$ may not be a relation at all.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620143-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
A $1$-ary relation on $A$ is a subset of what?
|
|
Back: $A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620149-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For what values of $n$ is an "$n$-ary relation on $A$" definitively a relation?
|
|
Back: $n > 1$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620155-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
For what values of $n$ is an "$n$-ary relation on $A$" *not* a "relation"?
|
|
Back: Potentially when $n = 1$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620160-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What *is* $n$ in term "$n$-ary relation on $A$"?
|
|
Back: A positive integer.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620165-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of "$n$-ary relations on $A$" and "relations" is more general?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620169-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Is $\{\langle x \rangle, \langle x, y \rangle, \langle x, y, z \rangle\}$ a relation?
|
|
Back: Indeterminate.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620173-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What must be true for $\{\langle x \rangle, \langle x, y \rangle, \langle x, y, z \rangle\}$ to be a relation?
|
|
Back: $x$ must be an ordered pair.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620178-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $\{\langle \varnothing \rangle, \langle \varnothing, \varnothing \rangle, \langle \varnothing, \varnothing, \varnothing \rangle\}$ a relation?
|
|
Back: $\langle \varnothing \rangle = \varnothing$ is not an ordered pair.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620182-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $\{\langle x, y \rangle, \langle x, y, z \rangle\}$ a relation?
|
|
Back: N/A. It is.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620187-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $x, y, z \in A$. Is $\{\langle x, y \rangle, \langle x, y, z \rangle\}$ a $2$-ary relation on $A$?
|
|
Back: No.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620193-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $x, y, z \in A$. *Why* isn't $\{\langle x, y \rangle, \langle x, y, z \rangle\}$ a $2$-ary relation on $A$?
|
|
Back: Because $\langle x, y, z \rangle \not\in A \times A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620199-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $x, y, z \in A$. Is $\{\langle x, y \rangle, \langle x, y, z \rangle\}$ a $3$-ary relation on $A$?
|
|
Back: No.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620203-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $x, y, z \in A$. *Why* isn't $\{\langle x, y \rangle, \langle x, y, z \rangle\}$ a $3$-ary relation on $A$?
|
|
Back: Because $\langle x, y \rangle \not\in (A \times A) \times A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620208-->
|
|
END%%
|
|
|
|
## Reflexivity
|
|
|
|
A relation $R$ is **reflexive on $A$** iff $xRx$ for all $x \in A$. In relational algebra, we define $R$ to be reflexive on $A$ iff $I_A \subseteq R$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is reflexivity of relation $R$ on set $A$ defined in FOL?
|
|
Back: $\forall x \in A, xRx$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721869969493-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is reflexivity of relation $R$ on set $A$ defined in relational algebra?
|
|
Back: $I_A \subseteq R$
|
|
Reference: “Equivalence Relation,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Equivalence_relation](https://en.wikipedia.org/w/index.php?title=Equivalence_relation&oldid=1235801091).
|
|
<!--ID: 1721869969498-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why is it incorrect to ask if $R$ is reflexive?
|
|
Back: We have to ask if $R$ is reflexive on some reference set.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429800-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $a$?
|
|
Back: N/A. We should ask if $R$ is reflexive on set $\{a\}$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429808-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ reflexive on $\{a\}$?
|
|
Back: N/A. It is.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429812-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ reflexive on $\{a, b\}$?
|
|
Back: Because $\langle b, b \rangle \not\in R$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429817-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, *why* isn't $R$ reflexive on $\{a, b\}$?
|
|
Back: Because $\langle b, b \rangle \not\in R$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429820-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
If $xRx$ for all $x \in A$, $R$ is said to be reflexive {on} $A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429824-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The term "reflexive" is used to describe what kind of mathematical object?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721693996250-->
|
|
END%%
|
|
|
|
### Irreflexivity
|
|
|
|
A relation $R$ is **irreflexive on $A$** iff $\neg xRx$ for all $x \in A$. That is, it is *never* the case that $xRx$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is irreflexivity of relation $R$ on set $A$ defined in FOL?
|
|
Back: $\forall x \in A, \neg xRx$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870888378-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why is it incorrect to ask if $R$ is irreflexive?
|
|
Back: We have to ask if $R$ is irreflexive on some reference set.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870888384-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ irreflexive on $a$?
|
|
Back: N/A. We should ask if $R$ is irreflexive on set $\{a\}$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870888391-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ irreflexive on $\{a\}$?
|
|
Back: Because $\langle a, a \rangle \in R$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870888395-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ irreflexive on $\{b\}$?
|
|
Back: N/A. It is.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870888400-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ irreflexive on $\{a, b\}$?
|
|
Back: Because $\langle a, a \rangle \in R$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870888406-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
If $\neg xRx$ for all $x \in A$, $R$ is said to be irreflexive {on} $A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870888411-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The term "irreflexive" is used to describe what kind of mathematical object?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870888417-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be neither reflexive nor irreflexive on the same set?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721911994966-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be both reflexive and irreflexive on the same set?
|
|
Back: No.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721911994996-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
If a nonempty relation isn't reflexive, is it irreflexive?
|
|
Back: Not necessarily.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721911995004-->
|
|
END%%
|
|
|
|
## Symmetry
|
|
|
|
A relation $R$ is **symmetric** iff whenever $xRy$, then $yRx$. In relational algebra, we define $R$ to be symmetric iff $R^{-1} \subseteq R$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is symmetry of relation $R$ defined in FOL?
|
|
Back: $\forall x, \forall y, xRy \Rightarrow yRx$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870204117-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is symmetry of relation $R$ defined in relational algebra?
|
|
Back: $R^{-1} \subseteq R$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870204123-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, b \rangle, \langle b, c \rangle\}$ symmetric?
|
|
Back: Because $aRb$ and $bRc$ but $\neg bRa$ and $\neg cRb$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429832-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, what additional member(s) must be added to make $R$ symmetric?
|
|
Back: $\langle b, a \rangle$ and $\langle c, b \rangle$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429835-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The term "symmetric" is used to describe what kind of mathematical object?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721694448727-->
|
|
END%%
|
|
|
|
### Antisymmetry
|
|
|
|
A relation $R$ is **antisymmetric** iff whenever $x \neq y$ and $xRy$, then $\neg yRx$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is antisymmetry of relation $R$ defined in FOL?
|
|
Back: $\forall x, \forall y, x \neq y \land xRy \Rightarrow \neg yRx$
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909725683-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
A relation $R$ on set $A$ that satisfies the following exhibits what property? $$\forall x, y \in A, xRy \land yRx \Rightarrow x = y$$
|
|
Back: Antisymmetry.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909971801-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle a, b \rangle, \langle b, a \rangle, \langle b, c \rangle\}$ antisymmetric?
|
|
Back: Because $aRb$ and $bRa$.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909725690-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle, \langle b, c \rangle\}$ antisymmetric?
|
|
Back: N/A. It is.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909725693-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be both reflexive and antisymmetric on the same set?
|
|
Back: Yes.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909725700-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be both symmetric and antisymmetric on the same set?
|
|
Back: Yes.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909725703-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be neither symmetric nor antisymmetric on the same set?
|
|
Back: Yes.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909971804-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of reflexivity, symmetry, and/or antisymmetry does $\{\langle a, a \rangle, \langle b, b \rangle\}$ exhibit?
|
|
Back: All three.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909725707-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of reflexivity, symmetry, and/or antisymmetry does $\{\langle a, a \rangle, \langle b, c \rangle\}$ exhibit?
|
|
Back: Antisymmetry.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909725711-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of reflexivity, symmetry, and/or antisymmetry does $\{\langle a, a \rangle, \langle b, c \rangle, \langle c, b \rangle\}$ exhibit?
|
|
Back: Symmetry.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909725715-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of reflexivity, symmetry, and/or antisymmetry does $\{\langle a, b \rangle, \langle b, c \rangle, \langle c, b \rangle\}$ exhibit?
|
|
Back: None of them.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721909971807-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
If a nonempty relation isn't symmetric, is it antisymmetric?
|
|
Back: Not necessarily.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721911804446-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The term "antisymmetric" is used to describe what kind of mathematical object?
|
|
Back: Relations.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721912048138-->
|
|
END%%
|
|
|
|
### Asymmetry
|
|
|
|
A relation $R$ is **asymmetric** iff whenever $xRy$, then $\neg yRx$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is antisymmetry of relation $R$ defined in FOL?
|
|
Back: $\forall x, \forall y, xRy \Rightarrow \neg yRx$
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What distinguishes the antecedent of antisymmetry's and asymmetry's FOL definition?
|
|
Back: The former only considers *distinct* pairs of elements.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949017-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Are antisymmetric relations necessarily asymmetric?
|
|
Back: No.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949023-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Are asymmetric relations necessarily antisymmetric?
|
|
Back: Yes.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949029-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
A relation is {asymmetric} if and only if it is both {irreflexive} and {antisymmetric}.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949033-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a relation be both symmetric and asymmetric?
|
|
Back: Yes.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949037-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Give an example of a relation that is both symmetric and asymmetric.
|
|
Back: $\varnothing$
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be both symmetric and asymmetric?
|
|
Back: No.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949042-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be neither symmetric nor asymmetric?
|
|
Back: Yes.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721911667937-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Give an example of a nonempty relation that is both symmetric and asymmetric.
|
|
Back: N/A.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1723245187584-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be both reflexive and asymmetric on the same set?
|
|
Back: No.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949047-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of reflexivity, symmetry, and/or asymmetry does $\{\langle a, a \rangle, \langle b, b \rangle\}$ exhibit?
|
|
Back: Reflexivity and symmetry.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949051-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of reflexivity, symmetry, and/or asymmetry does $\{\langle a, a \rangle, \langle b, c \rangle\}$ exhibit?
|
|
Back: None of them.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949055-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of reflexivity, symmetry, and/or asymmetry does $\{\langle a, b \rangle, \langle b, c \rangle\}$ exhibit?
|
|
Back: Asymmetry.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721910949059-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
A relation $R$ is asymmetric if and only if what other two properties of $R$ hold?
|
|
Back: $R$ is both irreflexive and antisymmetric.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721911011861-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
If a nonempty relation isn't symmetric, is it asymmetric?
|
|
Back: Not necessarily.
|
|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
|
|
<!--ID: 1721911804453-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The term "asymmetric" is used to describe what kind of mathematical object?
|
|
Back: Relations.
|
|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
|
|
<!--ID: 1721912048142-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
A relation $R$ is asymmetric iff $R$ is {antisymmetric} and {irreflexive}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187594-->
|
|
END%%
|
|
|
|
## Transitivity
|
|
|
|
A relation $R$ is **transitive** iff whenever $xRy$ and $yRz$, then $xRz$. In relational algebra, we define $R$ to be transitive iff $R \circ R \subseteq R$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is transitivity of relation $R$ defined in FOL?
|
|
Back: $\forall x, \forall y, \forall z, xRy \land yRz \Rightarrow xRz$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870318644-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is transitivity of relation $R$ defined in relational algebra?
|
|
Back: $R \circ R \subseteq R$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721870318654-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, b \rangle, \langle b, c \rangle\}$ transitive?
|
|
Back: Because $\langle a, c \rangle \not\in R$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720967429846-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of symmetric relations and transitive relations is more general?
|
|
Back: N/A.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1720969371859-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The term "transitive" is used to describe what kind of mathematical object?
|
|
Back: Relations.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1721694448736-->
|
|
END%%
|
|
|
|
## Connected
|
|
|
|
A binary relation $R$ on set $A$ is said to be **connected** if for any *distinct* $x, y \in A$, either $xRy$ or $yRx$. The relation is **strongly connected** if for *all* $x, y \in A$, either $xRy$ or $yRx$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is connectivity of relation $R$ on set $A$ defined in FOL?
|
|
Back: $\forall x, y \in A, x \neq y \Rightarrow xRy \lor yRx$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1722735199628-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, b \rangle\}$ connected on set $\{a, b\}$?
|
|
Back: N/A. It is.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1722735199637-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle\}$ connected on set $\{a, b\}$?
|
|
Back: Because $\langle a, b \rangle \not\in R$ and $\langle b, a \rangle \not\in R$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1722735199645-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle\}$ connected on set $\{a, b\}$?
|
|
Back: Because neither $aRb$ nor $bRa$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1722735199650-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of reflexivity or connectivity is the more general concept?
|
|
Back: N/A.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1722735199658-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What members must be added to make $R = \{\langle a, b \rangle, \langle b, c \rangle, \langle c, a \rangle\}$ connected on $\{a, b, c\}$?
|
|
Back: N/A.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1722735199662-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is strong connectivity of relation $R$ on set $A$ defined in FOL?
|
|
Back: $\forall x, y \in A, xRy \lor yRx$
|
|
Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
|
|
<!--ID: 1722735199672-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, b \rangle\}$ strongly connected on set $\{a, b\}$?
|
|
Back: Because $\neg aRa$ and $\neg bRb$.
|
|
Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
|
|
<!--ID: 1722735199683-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What members must be added to make $R = \{\langle a, b \rangle, \langle b, c \rangle, \langle c, a \rangle\}$ strongly connected on $\{a, b, c\}$?
|
|
Back: $\langle a, a \rangle$, $\langle b, b \rangle$, and $\langle c, c \rangle$.
|
|
Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
|
|
<!--ID: 1722735199688-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of strong connectivity or reflexivity is the more general concept?
|
|
Back: Reflexivity.
|
|
Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
|
|
<!--ID: 1722735199695-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
{1:Antisymmetry} is to {2:asymmetry} as {2:connectivity} is to {1:strong connectivity}.
|
|
Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
|
|
<!--ID: 1722735199702-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why might we say asymmetry is "strong antisymmetry"?
|
|
Back: Asymmetry implies antisymmetry.
|
|
Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
|
|
<!--ID: 1722735199707-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What makes "strong connectivity" stronger than "connectivity"?
|
|
Back: The former implies the latter.
|
|
Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
|
|
<!--ID: 1722735199715-->
|
|
END%%
|
|
|
|
## Trichotomy
|
|
|
|
A binary relation $R$ on $A$ is **trichotomous** if for all $x, y \in A$, exactly one of the following holds: $$xRy, \quad x = y, \quad yRx$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is trichotomy of relation $R$ on set $A$ defined in FOL?
|
|
Back: $\forall x, y \in A, (xRy \land x \neq y \land \neg yRx) \lor (\neg xRy \land x = y \land \neg yRx) \lor (\neg xRy \land x \neq y \land yRx)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187598-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle 2, 3 \rangle, \langle 2, 5 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
|
|
Back: N/A. It is.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187602-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle 2, 3 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
|
|
Back: Because no ordered pair relates $2$ and $5$ together.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187609-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* isn't $R = \{\langle a, a \rangle\}$ trichotomous on $\{a\}$?
|
|
Back: Because $aRa$ and $a = a$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187617-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a relation be both reflexive and trichotomous?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187621-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be both reflexive and trichotomous?
|
|
Back: No.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187628-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Can a nonempty relation be both irreflexive and trichotomous?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187633-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of trichotomy or irreflexivity is more general?
|
|
Back: Irreflexivity.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187638-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* must trichotomous relations on (say) set $A$ be irreflexive?
|
|
Back: For any $x \in A$, it follows $x = x$. Then $\neg xRx$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1723245187643-->
|
|
END%%
|
|
|
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%%ANKI
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Basic
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Can a nonempty relation be both symmetric and trichotomous?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1723245187648-->
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END%%
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%%ANKI
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Basic
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Can a nonempty relation be both antisymmetric and trichotomous?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1723245187654-->
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END%%
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%%ANKI
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Basic
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Which of antisymmetry or trichotomy is more general?
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Back: Antisymmetry.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1723245187659-->
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END%%
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%%ANKI
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Basic
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*Why* must trichotomous relations on (say) set $A$ be antisymmetric?
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Back: For any $x, y \in A$, if $x \neq y$ then $xRy$ or $yRx$ but not both.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1723245187664-->
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END%%
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%%ANKI
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Cloze
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A relation $R$ is trichotomous iff $R$ is {asymmetric} and {connected}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1723245187669-->
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END%%
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## Bibliography
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* “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
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* “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
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* “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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* “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
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* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |