20 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 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.
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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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Basic
Is \varnothing a relation?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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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 Rx \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).
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%%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).
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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).
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Basic
How is the ordering relation < on \{2, 3, 5\} visualized?
Back:
!
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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Basic
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}}
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{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).
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Basic
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).
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Basic
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}}
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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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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"?
Back: For each y \in \mathop{\text{ran}}A, there exists a unique x such that xAy.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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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.
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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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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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Bibliography
- “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).