notebook/notes/set/relations.md

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Relations Obsidian::STEM set::relation
relation
set

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.

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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).

END%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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).

END%%

%%ANKI 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%%

%%ANKI 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 R
  • x \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).

END%%

%%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%%

%%ANKI 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).

END%%

%%ANKI Basic How is the ordering relation < on \{2, 3, 5\} visualized? Back: !relation-ordering-example.png Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%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).

END%%

%%ANKI 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%%

%%ANKI 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).

END%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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).

END%%

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.

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

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.

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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:

  • R is reflexive on A if xRx for all x \in A.
  • R is symmetric if whenever xRy, then yRx.
  • R is transitive if whenever xRy and yRz, then xRz.

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%ANKI 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%%

%%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 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%%

%%ANKI 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%%

%%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 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%%

%%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%%

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].

%%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 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 relation. 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 {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).

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%%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).

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%%ANKI Basic Is A a partition of set A? Back: No. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic Is \{A\} a partition of set A? Back: Yes. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%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).

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%%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).

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%%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).

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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).

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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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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