599 lines
20 KiB
Markdown
599 lines
20 KiB
Markdown
---
|
|
title: Relations
|
|
TARGET DECK: Obsidian::STEM
|
|
FILE TAGS: set::relation
|
|
tags:
|
|
- 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).
|
|
<!--ID: 1717678753102-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1717679524930-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which of ordered pairs or sets is more general?
|
|
Back: Sets.
|
|
<!--ID: 1717678753108-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1717678753111-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1717678753116-->
|
|
END%%
|
|
|
|
%%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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
|
<!--ID: 1717678753120-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1717678753124-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1717678753129-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1717678753134-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1717678753139-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1717678753145-->
|
|
END%%
|
|
|
|
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).
|
|
<!--ID: 1718107987764-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987776-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987783-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the ordering relation $<$ on $\{2, 3, 5\}$ visualized?
|
|
Back:
|
|
![[relation-ordering-example.png]]
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718107987794-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987803-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987813-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987822-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987831-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987840-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987850-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987862-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the most general mathematical object the $\mathop{\text{dom}}$ operation can be applied to?
|
|
Back: Sets.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718546439334-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A$ be a set containing no ordered pairs. What is $\mathop{\text{dom}} A$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739893-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A = \{\{\{x\}, \{x, y\}\}, \{z\}\}$. What is $\mathop{\text{dom}} A$?
|
|
Back: $\{x\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739898-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For any set $\mathscr{A}$, $\mathop{\text{dom}}\bigcup\mathscr{A}$ {$=$} $\bigcup\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739901-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The following is analagous to what predicate logical expression of commuting quantifiers?$$\mathop{\text{dom}}\bigcup\mathscr{A} = \bigcup\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
|
|
Back: $\exists x, \exists y, P(x, y) \Leftrightarrow \exists y, \exists x, P(x, y)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739907-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For any set $\mathscr{A}$, $\mathop{\text{dom}}\bigcap\mathscr{A}$ {$\subseteq$} $\bigcap\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739910-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The following is analagous to what predicate logical expression of commuting quantifiers? $$\mathop{\text{dom}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
|
|
Back: $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739914-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For any set $\mathscr{A}$, $\mathop{\text{ran}}\bigcup\mathscr{A}$ {$=$} $\bigcup\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739918-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The following is analagous to what predicate logical expression of commuting quantifiers? $$\mathop{\text{ran}}\bigcup\mathscr{A} = \bigcup\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$$
|
|
Back: $\exists x, \exists y, P(x, y) \Leftrightarrow \exists y, \exists x, P(x, y)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739922-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
For any set $\mathscr{A}$, $\mathop{\text{ran}}\bigcap\mathscr{A}$ {$\subseteq$} $\bigcap\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739926-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The following is analagous to what predicate logical expression of commuting quantifiers? $$\mathop{\text{ran}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$$
|
|
Back: $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739931-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987872-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987880-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the most general mathematical object the $\mathop{\text{ran}}$ operation can be applied to?
|
|
Back: Sets.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718546439338-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A$ be a set containing no ordered pairs. What is $\mathop{\text{ran}} A$?
|
|
Back: $\varnothing$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739936-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A = \{\{\{x\}, \{x, y\}\}, \{z\}\}$. What is $\mathop{\text{ran}} A$?
|
|
Back: $\{y\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739940-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987887-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1718107987897-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A = \{\{\{x\}, \{x, y\}\}, \{z\}\}$. What is $\mathop{\text{fld}} A$?
|
|
Back: $\{x, y\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739945-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
If $\langle x, y \rangle \in A$, what sets are in $\bigcup A$?
|
|
Back: $\{x\}$ and $\{x, y\}$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739950-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
If $\langle x, y \rangle \in A$, what sets are in $\bigcup \bigcup A$?
|
|
Back: $x$ and $y$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739955-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the most general mathematical object the $\mathop{\text{fld}}$ operation can be applied to?
|
|
Back: Sets.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718546439341-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
$\mathop{\text{fld}} R = \bigcup \bigcup R$ is necessary for what condition?
|
|
Back: $R$ is a relation.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718327739961-->
|
|
END%%
|
|
|
|
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 $xAy$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does it mean for a set $A$ to be "single-valued"?
|
|
Back: For each $x \in \mathop{\text{dom}}A$, there exists a unique $y$ such that $xAy$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718427443355-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does it mean for a set $A$ to be "single-rooted"?
|
|
Back: For each $y \in \mathop{\text{ran}}A$, there exists a unique $x$ such that $xRy$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718465870483-->
|
|
END%%
|
|
|
|
## n-ary Relations
|
|
|
|
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 $\langle x \rangle = x$.
|
|
|
|
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.
|
|
|
|
%%ANKI
|
|
Basic
|
|
Ordered triple $\langle x, y, z \rangle$ is "syntactic sugar" for what?
|
|
Back: $\langle \langle x, y \rangle, z \rangle$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620058-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Ordered quadruple $\langle x_1, x_2, x_3, x_4 \rangle$ is "syntactic sugar" for what?
|
|
Back: $\langle \langle \langle x_1, x_2 \rangle, x_3 \rangle, x_4 \rangle$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620086-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
A $1$-tuple $\langle x \rangle$ is "syntactic sugar" for what?
|
|
Back: $x$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620091-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What simpler construct are $n$-tuples constructed from?
|
|
Back: Ordered pairs.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1718329620096-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Are $n$-tuples defined in a left- or right-associative way?
|
|
Back: Left-associative.
|
|
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 $A$" 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
|
|
Is $\{\langle x, y \rangle, \langle x, y, z \rangle\}$ a relation?
|
|
Back: Yes.
|
|
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%%
|
|
|
|
## Bibliography
|
|
|
|
* “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).
|
|
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |