37 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Order | Obsidian::STEM | set::order |
|
Overview
An order refers to a binary relations that defines how elements of a set relate to one another in terms of "less than", "equal to", or "greater than".
%%ANKI Cloze An order is a {2}-ary relation. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic In the context of order theory, what is an order? Back: A binary relation that defines how elements of a set relate to one another. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic In the context of order theory, what kind of mathematical object is an order? Back: A (binary) relation. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Preorders
A binary relation R on set A is a preorder on A iff it is reflexive on A and transitive.
%%ANKI
Basic
A binary relation on A is a preorder on A if it satisfies what two properties?
Back: Reflexivity on A and transitivity.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI Basic Which of preorders or equivalence relations are the more general concept? Back: Preorders. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI Basic Why are preorders named the way they are? Back: The name suggests its almost a partial order. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle\} a preorder on \{a\}?
Back: N/A. It is.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, b \rangle, \langle b, c \rangle, \langle a, c \rangle\} a preorder on \{a, b, c\}?
Back: Because R isn't reflexive on \{a, b, c\}.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, b \rangle \} a preorder on \{a, b\}?
Back: N/A. It is.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Cloze
Operator {\leq} typically denotes a {non-strict} preorder.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
A binary relation R on set A is a strict preorder on A iff it is irreflexive on A and transitive.
%%ANKI Basic What distinguishes a preorder from a strict preorder? Back: Strict preorders are irreflexive. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Basic
A binary relation on A is a strict preorder on A if it satisfies what two properties?
Back: Irreflexivity on A and transitivity.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI Basic What makes a strict preorder more strict than a non-strict preorder? Back: Strict preorders do not allow relating members to themselves. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle\} a strict preorder on \{a\}?
Back: R isn't irreflexive.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, b \rangle, \langle b, c \rangle, \langle a, c \rangle\} a strict preorder on \{a, b, c\}?
Back: N/A. It is.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, b \rangle \} a strict preorder on \{a, b\}?
Back: R isn't irreflexive.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI Cloze A {1:strict} preorder is equivalent to a {1:strict} partial order. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI Basic Why is a strict preorder also a strict partial order? Back: Irreflexivity and transitivity imply antisymmetry. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI Basic What equivalence in order theory serves as a mnemonic for "irreflexivity and transitivity imply asymmetry"? Back: A strict preorder is equivalent to a strict partial order. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI Basic Why can't a nonempty preorder be asymmetric? Back: Because reflexivity violates asymmetry. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
Cloze
Operator {<} typically denotes a {strict} preorder.
Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
Partial Orders
A binary relation R on set A is a partial order on A iff it is reflexive on A, antisymmetric, and transitive. In other words, a partial order is an antisymmetric preorder.
%%ANKI
Basic
A binary relation on A is a partial order on A if it satisfies what three properties?
Back: Reflexivity on A, antisymmetry, and transitivity.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Basic Which of preorders and partial orders is the more general concept? Back: Preorders. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Basic Which of partial orders and equivalence relations is the more general concept? Back: N/A. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Cloze A preorder satisfying {antisymmetry} is a {partial order}. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI Basic What two properties do partial orders and equivalence relations have in common? Back: Reflexivity and transitivity. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Basic What property distinguishes partial orders from equivalence relations? Back: The former is antisymmetric whereas the latter is symmetric. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Basic Why is a partial order named the way it is? Back: Not every pair of elements needs to be comparable. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Basic Can a relation be both an equivalence relation and a partial order? Back: Yes. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Basic Can a nonempty relation be both an equivalence relation and a partial order? Back: Yes. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, b \rangle\} a partial order on \{a, b\}?
Back: N/A. It is.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, c \rangle\} a partial order on \{a, b, c\}?
Back: It isn't reflexive on \{b, c\}.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, c \rangle, \langle c, b \rangle\} a partial order on \{a, b, c\}?
Back: It isn't reflexive on \{b, c\}, it isn't antisymmetric, and it isn't transitive.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
A binary relation R on set A is a strict partial order on A iff it is irreflexive on A, antisymmetric, and transitive.
%%ANKI Basic What distinguishes a partial order from a strict partial order? Back: Strict partial orders are irreflexive. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Basic
A binary relation on A is a strict partial order on A if it satisfies what three properties?
Back: Irreflexivity on A, antisymmetry, and transitivity.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Basic What makes a strict partial order more strict than a non-strict partial order? Back: Strict partial orders do not allow relating members to themselves. Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Cloze
Operator {<} typically denotes a {strict} partial order.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Cloze
Operator {\leq} typically denotes a {non-strict} partial order.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, c \rangle\} a strict partial order on \{a, b, c\}?
Back: Because it isn't irreflexive.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, c \rangle, \langle b, c \rangle\} a strict partial order on \{a, b, c\}?
Back: N/A. It is.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, b \rangle, \langle b, c \rangle, \langle c, b \rangle\} a strict partial order on \{a, b\}?
Back: It is neither antisymmetric nor transitive.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
Equivalence Relations
A binary relation R on set A is an equivalence relation on A iff it is reflexive on A, symmetric, and transitive. In other words, an equivalence relation is a symmetric preorder.
%%ANKI
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%%
%%ANKI
Basic
A binary relation on A is an equivalence relation on A if it satisfies what three properties?
Back: Reflexivity on A, symmetry, and transitivity.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Cloze A preorder satisfying {symmetry} is an {equivalence relation}. Reference: “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
END%%
%%ANKI
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
Why isn't R = \{\langle a, a \rangle\} an equivalence relation on \{a\}?
Back: N/A. It is.
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 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%%
%%ANKI
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%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, c \rangle\} an equivalence relation on \{a, b\}?
Back: It is neither reflexive on \{a, b\} nor symmetric.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Equivalence Classes
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 just write [x].
%%ANKI
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
How is set [x] defined?
Back: As \{t \mid xRt\} for some unspecified R.
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%%
%%ANKI
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%%
%%ANKI
Basic
What kind of mathematical object is R in [x]_R?
Back: A set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
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 x \in {2:\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). With maximum specificity, 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%%
%%ANKI
Basic
Given sets A and x, how can [x]_A be rewritten as an image?
Back: A[\![\{x\}]\!]
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given sets A and x, how can we write A[\![\{x\}]\!] more compactly?
Back: [x]_A
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%%
%%ANKI Basic What name is given to a member of a partition of a set? Back: A cell. Reference: John B. Fraleigh, A First Course in Abstract Algebra, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
END%%
%%ANKI
Basic
Let R be the equivalence relation induced by partition \Pi of A. What does A / R equal?
Back: \Pi.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let R be an equivalence relation on A. What equivalence relation does partition A / R induce?
Back: R.
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
Quotient set 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 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 A?
Back: A set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider set A / R. What kind of mathematical object is R?
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 set A / R defined?
Back: As \{[x]_R \mid x \in 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 \{[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 {1:R}) is an {2:equivalence class} whereas {2:A} modulo {2:R} is a {1:quotient set}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Total Order
A binary relation R on set A is a total order on A iff it is reflexive on A, antisymmetric, transitive, and strongly connected. In other words, a total order is a strongly connected partial order.
%%ANKI
Basic
A binary relation on A is a total order on A if it satisfies what four properties?
Back: Reflexivity on A, antisymmetry, transitivity, and strong connectivity.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI Basic Why is a total order named the way it is? Back: Every pair of elements needs to be comparable. Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI Basic Which of partial orders and total orders is the more general concept? Back: Partial orders. Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI Basic Which property of total orders is its name attributed to? Back: Strong connectivity. Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI Cloze A {total} order is a {partial} order satisfying {strong connectivity}. Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Cloze
Operator {\leq} typically denotes a {non-strict} total order.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, b \rangle\} a total order on \{a, b\}?
Back: It isn't strongly connected.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, a \rangle\} a total order on \{a, b\}?
Back: It is neither reflexive nor strongly connected.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle a, b \rangle, \langle b, b \rangle\} a total order on \{a, b\}?
Back: N/A. It is.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
A binary relation R on set A is a strict total order on A iff it is irreflexive on A, antisymmetric, transitive, and connected. In other words, a strict total order is a connected strict partial order.
%%ANKI
Basic
A binary relation on A is a strict total order on A if it satisfies what four properties?
Back: Irreflexivity on A, antisymmetry, transitivity, and connectivity.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Cloze
Operator {<} typically denotes a {strict} total order.
Reference: “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
END%%
%%ANKI Basic Which of strict total orders and strict partial orders is the more general concept? Back: Strict partial orders. Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI Cloze A {strict total} order is a {strict partial} order satisfying {connectivity}. Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI Cloze A {1:non-strict} total order satisfies {2:strong connectivity} whereas a {2:strict} total order satisfies {1:connectivity}. Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, b \rangle\} a strict total order on \{a, b\}?
Back: It is neither irreflexive nor connected.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, a \rangle, \langle b, a \rangle\} a strict total order on \{a, b\}?
Back: It isn't irreflexive.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, b \rangle\} a strict total order on \{a, b\}?
Back: N/A. It is.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
%%ANKI
Basic
Why isn't R = \{\langle a, b \rangle, \langle b, a \rangle\} a strict total order on \{a, b\}?
Back: It is neither antisymmetric nor transitive.
Reference: “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.
END%%
Bibliography
- “Equivalence Relation,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Equivalence_relation.
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
- John B. Fraleigh, A First Course in Abstract Algebra, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
- “Partially Ordered Set,” in Wikipedia, June 22, 2024, https://en.wikipedia.org/w/index.php?title=Partially_ordered_set.
- “Partition of a Set,” in Wikipedia, June 18, 2024, https://en.wikipedia.org/w/index.php?title=Partition_of_a_set.
- “Preorder,” in Wikipedia, July 21, 2024, https://en.wikipedia.org/w/index.php?title=Preorder.
- “Total Order.” In Wikipedia, April 9, 2024. https://en.wikipedia.org/w/index.php?title=Total_order.