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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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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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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Basic
Which of ordered pairs or sets is more general?
Back: Sets.
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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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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).
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).
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:
* $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$
The following is analagous to what logical expression of commuting quantifiers?$$\mathop{\text{dom}}\bigcup\mathscr{A} = \bigcup\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
The following is analagous to what logical expression of commuting quantifiers? $$\mathop{\text{dom}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
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).
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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).
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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).
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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).
The following is analagous to what logical expression of commuting quantifiers? $$\mathop{\text{ran}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$$
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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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).
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Basic
What does it mean for a set $A$ to be "single-rooted"?
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.
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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).
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Ordered quadruple $\langle x_1, x_2, x_3, x_4 \rangle$ is "syntactic sugar" for what?
Given relation $R$ and set $A$, $R$ is an **equivalence relation on $A$** iff $R$ is a binary relation on $A$ that is reflexive on $A$, symmetric, and transitive:
* $R$ is **reflexive on $A$** if $xRx$ for all $x \in A$.
* $R$ is **symmetric** if whenever $xRy$, then $yRx$.
* $R$ is **transitive** if whenever $xRy$ and $yRz$, then $xRz$.
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Cloze
Binary relation $R$ is {reflexive on $A$} iff {$xRx$ for all $x \in A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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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).
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Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive?
Back: N/A. The question must provide a reference set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $a$?
Back: N/A. We must ask if $R$ is reflexive on a set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $\{a\}$?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $\{a, b\}$?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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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).
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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).
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Cloze
Binary relation $R$ is {symmetric} iff {$xRy \Rightarrow yRx$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Basic
Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, is $R$ symmetric?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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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).
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Basic
Given $R = \{\langle a, a \rangle, \langle b, b \rangle\}$, which of reflexivity (on $\{a, b\}$), symmetry, and transitivity does $R$ exhibit?
Back: Reflexivity on $\{a, b\}$ and symmetry.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
* “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).
