---
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).
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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).
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END%%

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

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

Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$

%%ANKI
Basic
How is the Cartesian product of $A$ and $B$ denoted?
Back: $A \times B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397781-->
END%%

%%ANKI
Basic
Using ordered pairs, how is $A \times B$ defined?
Back: $\{\langle x, y \rangle \mid x \in A \land y \in B\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Who is attributed the representation of points in a plane?
Back: René Descartes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Why is the Cartesian product named the way it is?
Back: It is named after René Descartes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
Back: $\mathscr{P}\mathscr{P}A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Cloze
{$x \in A$} iff {$\{x\} \subseteq A$} iff {$\{x\} \in \mathscr{P}A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

## Bibliography

* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).