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:
* $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$
* “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).