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