diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 2190a91..0ed9a6c 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -168,7 +168,12 @@ "b-tree-initial.png", "b-tree-inserted-b.png", "b-tree-inserted-q.png", - "relation-ordering-example.png" + "relation-ordering-example.png", + "venn-diagram-union.png", + "venn-diagram-abs-comp.png", + "venn-diagram-intersection.png", + "venn-diagram-rel-comp.png", + "venn-diagram-symm-diff.png" ], "File Hashes": { "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8", @@ -348,7 +353,7 @@ "_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f", "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b", "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53", - "set/index.md": "b43367d200334c53d66fc85a876b9524", + "set/index.md": "7354797161e6720469801839b10b64c5", "set/graphs.md": "f0cd201673f2a999321dda6c726e8734", "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb", "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391", @@ -534,7 +539,7 @@ "_journal/2024-06/2024-06-04.md": "52b28035b9c91c9b14cef1154c1a0fa1", "_journal/2024-06-06.md": "3f9109925dea304e7172df39922cc95a", "_journal/2024-06/2024-06-05.md": "b06a0fa567bd81e3b593f7e1838f9de1", - "set/relations.md": "60fed0d4642d214767c077ca44983357", + "set/relations.md": "9323fc61ee983487e16f6dd4c1629957", "_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2", "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381", "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8", @@ -718,8 +723,9 @@ "_journal/2024-08/2024-08-14.md": "f7d1dede5ab6e4634ad9de3d3426c6f7", "_journal/2024-08-16.md": "a25c680684bcffc6a38cebbb448d9d97", "_journal/2024-08/2024-08-15.md": "7c3a96a25643b62b0064bf32cb17d92f", - "_journal/2024-08-17.md": "f9305a6a34de9a510260bed7d2bb1aca", - "_journal/2024-08/2024-08-16.md": "096d9147a9e3e7a947558f8dec763a2c" + "_journal/2024-08-17.md": "375318b50134c7cb45a8eb08bc7e586d", + "_journal/2024-08/2024-08-16.md": "096d9147a9e3e7a947558f8dec763a2c", + "set/order.md": "373f4336d4845a3c2188d2215ac5fbc4" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-08-17.md b/notes/_journal/2024-08-17.md index 656b2ef..0b602c6 100644 --- a/notes/_journal/2024-08-17.md +++ b/notes/_journal/2024-08-17.md @@ -8,4 +8,5 @@ title: "2024-08-17" - [ ] Sheet Music (10 min.) - [ ] Korean (Read 1 Story) -* Notes on strict preorders/partial orders. \ No newline at end of file +* Notes on strict preorders/partial [[order|orders]] and (strict) total orders. +* Finished chapter 3 "Relations and Functions". \ No newline at end of file diff --git a/notes/set/functions.md b/notes/set/functions.md index d20805a..648dc32 100644 --- a/notes/set/functions.md +++ b/notes/set/functions.md @@ -1309,6 +1309,12 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Cloze +Let $Q$, $A$, and $B$ be sets. Then {$Q \restriction (A - B)$} $=$ {$(Q \restriction A) - (Q \restriction B)$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). +END%% + %%ANKI Basic Consider sets $A$ and $B$. How is $B \restriction A$ rewritten as a composition? diff --git a/notes/set/index.md b/notes/set/index.md index af7c97a..c5a7f7e 100644 --- a/notes/set/index.md +++ b/notes/set/index.md @@ -999,6 +999,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +What does $\bigcap\,\{x\}$ evaluate to? +Back: $x$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Axiom of Choice This axiom assumes the existence of some choice function capable of selecting some element from a nonempty set. Note this axiom is controversial because it is non-constructive: there is no procedure we can follow to decide which element was chosen. diff --git a/notes/set/order.md b/notes/set/order.md new file mode 100644 index 0000000..43bc2a4 --- /dev/null +++ b/notes/set/order.md @@ -0,0 +1,965 @@ +--- +title: Order +TARGET DECK: Obsidian::STEM +FILE TAGS: set::order +tags: + - order + - set +--- + +## Overview + +An **order** refers to a binary [[relations|relation]] 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? +What is a preorder on $A$? +Back: Reflexivity on $A$ and transitivity. +Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +END%% + +%%ANKI +Cloze +Operator {$\leq$} typically denote a {non-strict} preorder. +Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +END%% + +%%ANKI +Basic +*Why* is a strict preorder also a strict partial order? +Back: Irreflexivity and transitivity imply asymmetry (and antisymmetry). +Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +END%% + +%%ANKI +Cloze +Operator {$<$} typically denote a {strict} preorder. +Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +END%% + +%%ANKI +Basic +*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ a 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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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 antisymmetric. +Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +END%% + +%%ANKI +Cloze +Operator {$<$} typically denote a {strict} partial order. +Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +END%% + +%%ANKI +Cloze +Operator {$\leq$} typically denote 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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). + +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 isn't 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 $R$). 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: “Partition of a Set,” in _Wikipedia_, June 18, 2024, [https://en.wikipedia.org/w/index.php?title=Partition_of_a_set](https://en.wikipedia.org/w/index.php?title=Partition_of_a_set&oldid=1229656401). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +END%% + +%%ANKI +Cloze +Operator {$\leq$} typically denote a {non-strict} total order. +Reference: “Total Order.” In _Wikipedia_, April 9, 2024. [https://en.wikipedia.org/w/index.php?title=Total_order](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +END%% + +%%ANKI +Cloze +Operator {$<$} typically denote a {strict} total order. +Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +END%% + +%%ANKI +Cloze +A {non-strict} total order satisfies {strong connectivity} whereas a {strict} total order satisfies {connectivity}. +Reference: “Total Order.” In _Wikipedia_, April 9, 2024. [https://en.wikipedia.org/w/index.php?title=Total_order](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +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](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). + +END%% + +## Bibliography + +* “Equivalence Relation,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Equivalence_relation](https://en.wikipedia.org/w/index.php?title=Equivalence_relation&oldid=1235801091). +* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). +* “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). +* “Partition of a Set,” in _Wikipedia_, June 18, 2024, [https://en.wikipedia.org/w/index.php?title=Partition_of_a_set](https://en.wikipedia.org/w/index.php?title=Partition_of_a_set&oldid=1229656401). +* “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). +* “Total Order.” In _Wikipedia_, April 9, 2024. [https://en.wikipedia.org/w/index.php?title=Total_order](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468). \ No newline at end of file diff --git a/notes/set/relations.md b/notes/set/relations.md index ad4b12e..4da550c 100644 --- a/notes/set/relations.md +++ b/notes/set/relations.md @@ -636,14 +636,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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 $a$? @@ -711,14 +703,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% -%%ANKI -Basic -Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ irreflexive? -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 $R$ irreflexive on $a$? @@ -1351,803 +1335,10 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% -## Preorders - -$R$ is a **preorder on $A$** iff $R$ is a binary relation that is reflexive on set $A$ and transitive. - -%%ANKI -Basic -What is a preorder on $A$? -Back: A binary relation on $A$ that is reflexive on $A$ and transitive. -Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -END%% - -%%ANKI -Basic -*Why* isn't $R = \{\langle a, a \rangle\}$ a preorder? -Back: N/A. The question must provide a reference set. -Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -END%% - -A **strict preorder** replaces reflexivity with irreflexivity. That is, $R$ is a strict preorder on $A$ iff $R$ is a binary relation on set $A$ that 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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -END%% - -%%ANKI -Basic -What is a strict preorder on $A$? -Back: A binary relation on $A$ that is irreflexive on $A$ and transitive. -Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -END%% - -%%ANKI -Basic -*Why* is a strict preorder also a strict partial order? -Back: Irreflexivity and transitivity imply asymmetry (and antisymmetry). -Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -END%% - -## Partial Orders - -$R$ is a **partial order on $A$** iff $R$ is a binary relation on set $A$ that is reflexive on $A$, antisymmetric, and transitive. In other words, a partial order is an antisymmetric preorder. - -%%ANKI -Basic -What is a partial order on $A$? -Back: A binary relation on $A$ that is reflexive on $A$, antisymmetric, and transitive. -Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -END%% - -%%ANKI -Basic -*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle\}$ a partial order? -Back: N/A. The question must provide a reference set. -Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -END%% - -%%ANKI -Basic -*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ a 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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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 antisymmetric. -Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -END%% - -A **strict partial order** replaces reflexivity with irreflexivity. That is, $R$ is a strict partial order on $A$ iff $R$ is a binary relation on set $A$ that 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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -END%% - -%%ANKI -Basic -What is a strict partial order on $A$? -Back: A binary relation on $A$ that is irreflexive on $A$, antisymmetric, and transitive. -Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -END%% - -%%ANKI -Cloze -Operator {$<$} typically denote a {strict} partial order. -Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -END%% - -%%ANKI -Cloze -Operator {$\leq$} typically denote 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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -END%% - -%%ANKI -Basic -*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle\}$ a strict partial order? -Back: N/A. The question must provide a reference set. -Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -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](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). - -END%% - -## Equivalence Relations - -$R$ is an **equivalence relation on $A$** iff $R$ is a binary relation on set $A$ that 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 -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 -A preorder satisfying {symmetry} is an {equivalence relation}. -Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). - -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? -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 -*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 isn't 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 $R$). 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: “Partition of a Set,” in _Wikipedia_, June 18, 2024, [https://en.wikipedia.org/w/index.php?title=Partition_of_a_set](https://en.wikipedia.org/w/index.php?title=Partition_of_a_set&oldid=1229656401). - -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%% - ## Bibliography * “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107). * “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822). * “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). * “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201). -* “Equivalence Relation,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Equivalence_relation](https://en.wikipedia.org/w/index.php?title=Equivalence_relation&oldid=1235801091). -* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). -* “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839). -* “Partition of a Set,” in _Wikipedia_, June 18, 2024, [https://en.wikipedia.org/w/index.php?title=Partition_of_a_set](https://en.wikipedia.org/w/index.php?title=Partition_of_a_set&oldid=1229656401). -* “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474). \ No newline at end of file +* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). \ No newline at end of file