36 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).
END%%
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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).
END%%
%%ANKI Basic Which of ordered pairs or sets is more general? Back: Sets.
END%%
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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).
END%%
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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.
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%%
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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).
END%%
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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).
END%%
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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).
END%%
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Basic
Is \varnothing a relation?
Back: Yes.
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 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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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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Cloze
For any binary relation R, R \subseteq {1:\mathop{\text{dom} }R} \times {1:\mathop{\text{ran} }R} \subseteq {2:\mathop{\text{fld} }R} \times {2:\mathop{\text{fld} }R}.
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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
Equivalence Relations
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:
Ris reflexive onAifxRxfor allx \in A.Ris symmetric if wheneverxRy, thenyRx.Ris transitive if wheneverxRyandyRz, thenxRz.
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
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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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
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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).
END%%
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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).
END%%
%%ANKI
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).
END%%
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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).
END%%
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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).
END%%
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Cloze
Binary relation R is {transitive} iff {xRy \land yRz \Rightarrow xRz}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given R = \{\langle a, b \rangle, \langle b, c \rangle\}, is R transitive?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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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 transitive?
Back: Just \langle a, c \rangle.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is an equivalence relation on A?
Back: A binary relation on A that is reflexive on A, symmetric, and transitive.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
An equivalence relation on A is a {2}-ary relation on A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given R = \{\langle a, a \rangle\}, is R an equivalence relation?
Back: N/A. The question must provide a reference set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given R = \{\langle a, a \rangle\}, is R an equivalence relation on \{a\}?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given R = \{\langle a, a \rangle, \langle b, c \rangle\}, is R an equivalence relation on \{a\}?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, c \rangle\} an equivalence relation on \{a\}?
Back: R is neither symmetric nor transitive.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic Which of symmetric relations and transitive relations is more general? Back: N/A. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Which of equivalence relations on A and symmetric relations is more general?
Back: Symmetric relations.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Which of binary relations on A and equivalence relations on A is more general?
Back: Binary relations on A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
The set [x]_R is defined by [x]_R = \{t \mid xRt\}. If R is an equivalence relation and x \in \mathop{\text{fld}}R, then [x]_R is called the equivalence class of x (modulo R). If the relation R is fixed by the context, we may write just [x].
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Basic
How is set [x]_R defined?
Back: As \{t \mid xRt\}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What is an equivalence class? Back: A set of members mutually related w.r.t an equivalence relation. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What kind of mathematical object is x in [x]_R?
Back: A set (or urelement).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What kind of mathematical object is R in [x]_R?
Back: A relation.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What compact notation is used to denote \{t \mid xRt\}?
Back: [x]_R
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
If {1:R is an equivalence relation} and {1:x \in \mathop{\text{fld} }R}, then [x]_R is called the {2:equivalence class of x} (modulo {2:R}).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider an equivalence class of x (modulo R). What kind of mathematical object is x?
Back: A set (or urelement).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider an equivalence class of x (modulo R). What kind of mathematical object is R?
Back: A relation.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider an equivalence class of x (modulo R). What condition does x necessarily satisfy?
Back: x \in \mathop{\text{fld}}R
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider an equivalence class of x (modulo R). What condition does R necessarily satisfy?
Back: R is an equivalence relation.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Assume R is an equivalence relation on A and that x, y \in A. Then {1:[x]_R} = {1:[y]_R} iff {2:xRy}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Partitions
A partition \Pi of a set A is a set of nonempty subsets of A that is disjoint and exhaustive.
%%ANKI Basic What kind of mathematical object is a partition of a set? Back: A set. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is a partition of a set A?
Back: A set of nonempty subsets of A that is disjoint and exhaustive.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \Pi be a partition of a set A. When does \Pi = \varnothing?
Back: If and only if A = \varnothing.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \Pi be a partition of set A. What property must each individual member of \Pi exhibit?
Back: Each member is nonempty.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \Pi be a partition of set A. What property must each pair of members of \Pi exhibit?
Back: Each pair must be disjoint.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let \Pi be a partition of set A. Which property do all the members of \Pi exhibit together?
Back: The members of \Pi must be exhaustive.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does it mean for a partition \Pi of A to be exhaustive?
Back: Every member of A must appear in one of the members of \Pi.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Is A a partition of set A?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Is \{A\} a partition of set A?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let A = \{1, 2, 3, 4\}. Why isn't \{\{1, 2\}, \{2, 3, 4\}\} a partition of A?
Back: Each pair of members of a partition of A must be disjoint.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let A = \{1, 2, 3, 4\}. Why isn't \{\{1\}, \{2\}, \{3\}\} a partition of A?
Back: The members of a partition of A must be exhaustive.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let A = \{1, 2, 3, 4\}. Why isn't \{\{1, 2, 3\}, \{4\}\} a partition of A?
Back: N/A. It is.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Assume \Pi is a partition of set A. Then the relation R is an equivalence relation: $xRy \Leftrightarrow (\exists B \in \Pi, x \in B \land y \in B)$
%%ANKI
Basic
Let \Pi be a partition of A. What equivalence relation R is induced?
Back: R such that xRy \Leftrightarrow (\exists B \in \Pi, x \in B \land y \in B)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Quotient Sets
If R is an equivalence relation on A, then the quotient set "A modulo R" is defined as $A / R = \{[x]_R \mid x \in A\}.$
The natural map (or canonical map) \phi : A \rightarrow A / R is given by \phi(x) = [x]_R.
Note that A / R, the set of all equivalence classes, is a partition of A.
%%ANKI
Basic
Let R be an equivalence relation on A. What partition is induced?
Back: A / R = \{[x]_R \mid x \in A\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Members of A / R are called what?
Back: Equivalence classes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
A / R is a partition of what set?
Back: A
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is quotient set A / R pronounced?
Back: As "A modulo R".
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider quotient set A / R. What kind of mathematical object is A?
Back: A set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider quotient set A / R. What kind of mathematical object is R?
Back: An equivalence relation on A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is quotient set A / R defined?
Back: As set \{[x]_R \mid x \in A\}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given quotient set A / R, what is the domain of its natural map?
Back: A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given quotient set A / R, what is the codomain of its natural map?
Back: A / R.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider quotient set A / R. How is the natural map \phi defined?
Back: \phi \colon A \rightarrow A / R given by \phi(x) = [x]_R.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given quotient set A / R, what is the domain of its canonical map?
Back: A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given quotient set A / R, what is the codomain of its canonical map?
Back: A / R.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider quotient set A / R. How is the canonical map \phi defined?
Back: \phi \colon A \rightarrow A / R given by \phi(x) = [x]_R.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider set \omega and equivalence relation \sim. How is the relevant quotient set denoted?
Back: As \omega / {\sim}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Let R be an equivalence relation on A and x \in A. Then {1:x (modulo R)} is an {2:equivalence class} whereas {2:A modulo R} is a {1:quotient set}.
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.
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).