4.3 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Relations | Obsidian::STEM | set::relation |
|
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.
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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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%%ANKI 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).
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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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%%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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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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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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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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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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Given two sets A and B, the Cartesian product A \times B is defined as: $A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$
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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).
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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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%%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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%%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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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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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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Bibliography
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).