Additional notes on equivalence relations.

c-declarations
Joshua Potter 2024-07-14 09:03:36 -06:00
parent 4921298a57
commit 2e77d20f15
3 changed files with 199 additions and 3 deletions

View File

@ -509,7 +509,7 @@
"_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": "63cdf623940e4076d127440574f6b6a2",
"set/relations.md": "b98d316d6e7595c5406352541cc7f9ee",
"_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2",
"_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381",
"_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8",
@ -601,7 +601,7 @@
"ontology/index.md": "c523aa6652b285a0f1c053cb77be6f85",
"ontology/permissivism.md": "5b66dd065aa66d5a2624eda032d75b94",
"ontology/properties.md": "d417db0cecf11b1ed2e17f165d879fa5",
"_journal/2024-07-14.md": "d790f7f6b13285b91cd972d927c20f96",
"_journal/2024-07-14.md": "3e04720086dca16f1c6b7412ec1706fd",
"_journal/2024-07/2024-07-13.md": "60e8eb09812660a2f2bf86ffafab5714"
},
"fields_dict": {

View File

@ -8,4 +8,5 @@ title: "2024-07-14"
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Notes on infinite Cartesian products and their relation to the axiom of choice.
* Notes on [[set#Cartesian Product|infinite Cartesian products]] and their relation to the [[set/index#Infinite Cartesian Product Form|axiom of choice]].
* Initial notes on [[relations#Equivalence Relations|equivalence relations]].

View File

@ -601,6 +601,201 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1718329620208-->
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).
<!--ID: 1720967429790-->
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).
<!--ID: 1720967429800-->
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).
<!--ID: 1720967429804-->
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).
<!--ID: 1720967429808-->
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).
<!--ID: 1720967429812-->
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).
<!--ID: 1720967429817-->
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).
<!--ID: 1720967429820-->
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).
<!--ID: 1720967429824-->
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).
<!--ID: 1720967429828-->
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).
<!--ID: 1720967429832-->
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).
<!--ID: 1720967429835-->
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).
<!--ID: 1720967429839-->
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).
<!--ID: 1720967429843-->
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).
<!--ID: 1720967429846-->
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).
<!--ID: 1720967429850-->
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).
<!--ID: 1720967429853-->
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).
<!--ID: 1720967429857-->
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).
<!--ID: 1720967429860-->
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).
<!--ID: 1720967429864-->
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).
<!--ID: 1720967429868-->
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).
<!--ID: 1720967429873-->
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).
<!--ID: 1720969371859-->
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).
<!--ID: 1720969371866-->
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).
<!--ID: 1720969371869-->
END%%
## Bibliography
* “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).