Notes on quotient sets, function kernels, and fibers.
parent
77138cec45
commit
af53475ee2
notes
.obsidian/plugins/obsidian-to-anki-plugin
_journal
algebra
algorithms
encoding
hashing
lambda-calculus
logic
ontology
|
@ -143,7 +143,8 @@
|
|||
"function-surjective.png",
|
||||
"function-general.png",
|
||||
"church-rosser.png",
|
||||
"infinite-cartesian-product.png"
|
||||
"infinite-cartesian-product.png",
|
||||
"function-kernel.png"
|
||||
],
|
||||
"File Hashes": {
|
||||
"algorithms/index.md": "3ac071354e55242919cc574eb43de6f8",
|
||||
|
@ -187,7 +188,7 @@
|
|||
"algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
|
||||
"algorithms/loop-invariant.md": "3b390e720f3b2a98e611b49a0bb1f5a9",
|
||||
"algorithms/running-time.md": "5efc0791097d2c996f931c9046c95f65",
|
||||
"algorithms/order-growth.md": "dd241870e1cfa0d43179a46213d5ed9c",
|
||||
"algorithms/order-growth.md": "8f6f38331bc4f7640f71794dd616bd23",
|
||||
"_journal/2024-02-08.md": "19092bdfe378f31e2774f20d6afbfbac",
|
||||
"algorithms/sorting/selection-sort.md": "73415c44d6f4429f43c366078fd4bf98",
|
||||
"algorithms/index 1.md": "6fada1f3d5d3af64687719eb465a5b97",
|
||||
|
@ -218,7 +219,7 @@
|
|||
"encoding/ascii.md": "34350e7b5a4109bcd21f9f411fda0dbe",
|
||||
"encoding/index.md": "071cfa6a5152efeda127b684f420d438",
|
||||
"c/strings.md": "aba6e449906d05aee98e3e536eb43742",
|
||||
"logic/truth-tables.md": "b00bf6d31f34bc2cae692642f823c8e1",
|
||||
"logic/truth-tables.md": "a9fe98af6cabc0e4b087086075e09af5",
|
||||
"logic/short-circuit.md": "a3fb33603a38a6d3b268556dcbdfa797",
|
||||
"logic/boolean-algebra.md": "56d2e0be2853d49b5dface7fa2d785a9",
|
||||
"_journal/2024-02-13.md": "6242ed4fecabf95df6b45d892fee8eb0",
|
||||
|
@ -274,7 +275,7 @@
|
|||
"filesystems/cas.md": "d41c0d2e943adecbadd10a03fd1e4274",
|
||||
"git/objects.md": "4ad7a2ab275b5573055ea0433be1e4d7",
|
||||
"git/index.md": "ca842957bda479dfa1170ae85f2f37b8",
|
||||
"encoding/integer.md": "ab0db8d48734867d42279fb2f2362d25",
|
||||
"encoding/integer.md": "f9786eab7f64ec63272dcca010961fe8",
|
||||
"_journal/2024-02-29.md": "f610f3caed659c1de3eed5f226cab508",
|
||||
"_journal/2024-02/2024-02-28.md": "7489377c014a2ff3c535d581961b5b82",
|
||||
"_journal/2024-03-01.md": "a532486279190b0c12954966cbf8c3fe",
|
||||
|
@ -461,7 +462,7 @@
|
|||
"_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c",
|
||||
"_journal/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c",
|
||||
"hashing/direct-addressing.md": "f75cc22e74ae974fe4f568a2ee9f951f",
|
||||
"hashing/index.md": "b643f6823777e4974e8d2c27255d975f",
|
||||
"hashing/index.md": "e3ab1dd55eb7bb97a73b48241a006deb",
|
||||
"set/classes.md": "6776b4dc415021e0ef60b323b5c2d436",
|
||||
"_journal/2024-05-19.md": "fddd90fae08fab9bd83b0ef5d362c93a",
|
||||
"_journal/2024-05/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c",
|
||||
|
@ -484,7 +485,7 @@
|
|||
"_journal/2024-05/2024-05-25.md": "3e8a0061fa58a6e5c48d12800d1ab869",
|
||||
"_journal/2024-05-27.md": "b36636d10eab34380f17f288868df3ae",
|
||||
"_journal/2024-05/2024-05-26.md": "abe84b5beae74baa25501c818e64fc95",
|
||||
"algebra/set.md": "e88847f21b467e7d243ac3d5941a75a0",
|
||||
"algebra/set.md": "ecf6aef8bc64fc14a73178adcdd3594e",
|
||||
"algebra/boolean.md": "ee41e624f4d3d3aca00020d9a9ae42c8",
|
||||
"git/merge-conflicts.md": "761ad6137ec51d3877f7d5b3615ca5cb",
|
||||
"_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337",
|
||||
|
@ -509,7 +510,7 @@
|
|||
"_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": "1daa000a3b57d4335783cf4fd759c746",
|
||||
"set/relations.md": "2750a1f7f82dfd146779c02572f8bfe9",
|
||||
"_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2",
|
||||
"_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381",
|
||||
"_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8",
|
||||
|
@ -535,10 +536,10 @@
|
|||
"_journal/2024-06/2024-06-12.md": "f82dfa74d0def8c3179d3d076f94558e",
|
||||
"_journal/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307",
|
||||
"_journal/2024-06/2024-06-13.md": "e2722a00585d94794a089e8035e05728",
|
||||
"set/functions.md": "cd38f47de7e4ecf3a55434865efa6877",
|
||||
"set/functions.md": "b41c04a596a7e711801c32eff9333a3e",
|
||||
"_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4",
|
||||
"_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307",
|
||||
"lambda-calculus/beta-reduction.md": "e233c8352a8180d19f7b717946c379d1",
|
||||
"lambda-calculus/beta-reduction.md": "e40e64be380bbe7852cfe6a310e400bf",
|
||||
"_journal/2024-06-16.md": "ded6ab660ecc7c3dce3afd2e88e5a725",
|
||||
"_journal/2024-06/2024-06-15.md": "c3a55549da9dfc2770bfcf403bf5b30b",
|
||||
"_journal/2024-06-17.md": "63df6757bb3384e45093bf2b9456ffac",
|
||||
|
@ -598,11 +599,18 @@
|
|||
"_journal/2024-07/2024-07-12.md": "6603ed8a3f9a9e87bf40e81b03e96356",
|
||||
"hashing/static.md": "3ec6eaee73fb9b599700f5a56b300b83",
|
||||
"hashing/addressing.md": "a78c0cbea13bc9deeadb2fc643c122ce",
|
||||
"ontology/index.md": "13e47f12ae5cf9816165c3e4f4090c1f",
|
||||
"ontology/index.md": "15e97e3e8068660314499fb4d1bdd53e",
|
||||
"ontology/permissivism.md": "5b66dd065aa66d5a2624eda032d75b94",
|
||||
"ontology/properties.md": "d417db0cecf11b1ed2e17f165d879fa5",
|
||||
"_journal/2024-07-14.md": "9a74d2dd0f44db58e14f57c8908c3342",
|
||||
"_journal/2024-07/2024-07-13.md": "60e8eb09812660a2f2bf86ffafab5714"
|
||||
"_journal/2024-07/2024-07-13.md": "60e8eb09812660a2f2bf86ffafab5714",
|
||||
"_journal/2024-07-15.md": "462fb4294cbbe8855071c638351df147",
|
||||
"_journal/2024-07/2024-07-14.md": "c4666b502d97387e05fb77c4139cae23",
|
||||
"_journal/2024-07-16.md": "0f3832a9afc331597e626864f24d6498",
|
||||
"_journal/2024-07/2024-07-15.md": "462fb4294cbbe8855071c638351df147",
|
||||
"ontology/nominalism.md": "46245c644238157e15c7cb6def27d90a",
|
||||
"_journal/2024-07-17.md": "e0371a91e99f131e7258cc82c2a04cc8",
|
||||
"_journal/2024-07/2024-07-16.md": "149222eab7a7f58993b8e4dc8a3fb884"
|
||||
},
|
||||
"fields_dict": {
|
||||
"Basic": [
|
||||
|
|
|
@ -0,0 +1,11 @@
|
|||
---
|
||||
title: "2024-07-17"
|
||||
---
|
||||
|
||||
- [x] Anki Flashcards
|
||||
- [x] KoL
|
||||
- [x] OGS
|
||||
- [ ] Sheet Music (10 min.)
|
||||
- [ ] Korean (Read 1 Story)
|
||||
|
||||
* Notes on [[relations#Quotient Sets|quotient sets]], function [[functions#Kernels|kernels]], and fibers.
|
|
@ -11,4 +11,5 @@ title: "2024-07-14"
|
|||
* 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]].
|
||||
* Read chapter 2 "How to Raise Money" in "Venture Deals".
|
||||
* Finished another read of "A Cardinal Worry for Permissive Metaontology".
|
||||
* Finished another read of "A Cardinal Worry for Permissive Metaontology".
|
||||
* Watched [Lecture 1 "Introduction to Ontology"](https://www.youtube.com/watch?v=9AsRE437e7I).
|
|
@ -0,0 +1,11 @@
|
|||
---
|
||||
title: "2024-07-15"
|
||||
---
|
||||
|
||||
- [x] Anki Flashcards
|
||||
- [x] KoL
|
||||
- [x] OGS
|
||||
- [ ] Sheet Music (10 min.)
|
||||
- [ ] Korean (Read 1 Story)
|
||||
|
||||
* Notes on [[relations#Partitions|partitions]] and equivalence classes.
|
|
@ -0,0 +1,11 @@
|
|||
---
|
||||
title: "2024-07-16"
|
||||
---
|
||||
|
||||
- [x] Anki Flashcards
|
||||
- [x] KoL
|
||||
- [ ] OGS
|
||||
- [ ] Sheet Music (10 min.)
|
||||
- [ ] Korean (Read 1 Story)
|
||||
|
||||
* Brief notes on [[nominalism]].
|
|
@ -93,7 +93,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
|
||||
Suppose $x, y \in A$. What set, derived from $A$, is $\langle x, y \rangle$ a member of?
|
||||
Back: $\mathscr{P}\mathscr{P}A$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1717679397848-->
|
||||
|
@ -758,8 +758,8 @@ Let $A$, $B$, and $C$ be arbitrary sets. Then
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
What kind of propositional logical statement are the monotonicity properties of $\subseteq$?
|
||||
Back: An implication.
|
||||
The monotonicity properties of $\subseteq$ are what kind of propositional logical statement?
|
||||
Back: Implications.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1717073536967-->
|
||||
END%%
|
||||
|
|
|
@ -690,7 +690,7 @@ END%%
|
|||
%%ANKI
|
||||
Basic
|
||||
What names are usually given to the existentially quantified identifers in $o(g(n))$'s definition?
|
||||
Back: $n_0$.
|
||||
Back: $n_0$
|
||||
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
||||
<!--ID: 1709519002328-->
|
||||
END%%
|
||||
|
@ -1035,7 +1035,7 @@ END%%
|
|||
%%ANKI
|
||||
Basic
|
||||
What is the symmetric property of $\Omega$-notation?
|
||||
Back: N/A
|
||||
Back: N/A.
|
||||
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
||||
<!--ID: 1709752223486-->
|
||||
END%%
|
||||
|
|
|
@ -557,7 +557,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
What is the precise definition of the two's-complement of a $w$-bit number?
|
||||
What is the precise definition of the two's-complement of a $w$-bit number $x$?
|
||||
Back: The complement of $x$ with respect to $2^w$.
|
||||
Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
|
||||
<!--ID: 1709060837145-->
|
||||
|
|
|
@ -404,7 +404,7 @@ END%%
|
|||
%%ANKI
|
||||
Basic
|
||||
Let $h$ be a division method hash function. What does $h(10)$ evaluate to?
|
||||
Back: $10 \bmod{m}$ where $m$ is the number of slots in the hash table.
|
||||
Back: To $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
|
||||
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
||||
Tags: hashing::static
|
||||
<!--ID: 1720889385419-->
|
||||
|
@ -428,6 +428,14 @@ Tags: hashing::static
|
|||
<!--ID: 1720889385429-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Why does the division method prefer a prime number of slots?
|
||||
Back: To operate as independently as possible of the input keys.
|
||||
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
||||
<!--ID: 1721218408542-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider hash function $h(k) = k \bmod{m}$. What method was likely used to produce this?
|
||||
|
|
|
@ -567,7 +567,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
What does the Church-Rosser theorem state in terms of confluence?
|
||||
What does the Church-Rosser theorem for $\triangleright_\beta$ state in terms of confluence?
|
||||
Back: $\beta$-reduction is confluent.
|
||||
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
|
||||
<!--ID: 1719577152613-->
|
||||
|
|
|
@ -50,7 +50,7 @@ $$
|
|||
%%ANKI
|
||||
Basic
|
||||
What construct is used to prove every proposition can be written in DNF or CNF?
|
||||
Back: Truth tables
|
||||
Back: Truth tables.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1707311868994-->
|
||||
END%%
|
||||
|
|
|
@ -21,7 +21,7 @@ END%%
|
|||
%%ANKI
|
||||
Basic
|
||||
Who is attributed *the* ontological question?
|
||||
Back: Quine.
|
||||
Back: Willard Van Orman Quine.
|
||||
Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009).
|
||||
<!--ID: 1720912259767-->
|
||||
END%%
|
||||
|
|
|
@ -0,0 +1,47 @@
|
|||
---
|
||||
title: Nominalism
|
||||
TARGET DECK: Obsidian::H&SS
|
||||
FILE TAGS: ontology::nominalism
|
||||
tags:
|
||||
- nominalism
|
||||
- ontology
|
||||
---
|
||||
|
||||
## Overview
|
||||
|
||||
**Anti-realists** about a category are those who don't believe entities of said category exist. **Realists** about a category are those that do. **Nominalism** refers to the stance that no abstract objects exist. That is, nominalists are anti-realists in regards to abstract entities.
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What does it mean for a person to be anti-realist about category $X$?
|
||||
Back: They do not believe entities of $X$ exist.
|
||||
Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
|
||||
<!--ID: 1721137271035-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What does it mean for a person to be realist about category $X$?
|
||||
Back: They believe entities belonging to $X$ exist.
|
||||
Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
|
||||
<!--ID: 1721137271065-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does Effingham define nominalism?
|
||||
Back: As anti-realism towards abstracta.
|
||||
Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
|
||||
<!--ID: 1721137271072-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Cloze
|
||||
Roughly speaking, {1:permissivism} is to {2:realism} whereas {2:nominalism} is to {1:anti-realism}.
|
||||
Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
|
||||
<!--ID: 1721137271080-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
* Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
|
|
@ -1327,7 +1327,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider sets $A$ and $B$. How is $A \cap B$ be rewritten as a function under some image?
|
||||
Consider sets $A$ and $B$. How is $A \cap B$ rewritten as a function under some image?
|
||||
Back: $I_A[\![B]\!]$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720885546358-->
|
||||
|
@ -1634,7 +1634,174 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
|
|||
<!--ID: 1720819771087-->
|
||||
END%%
|
||||
|
||||
## Kernels
|
||||
|
||||
Let $F \colon A \rightarrow B$. Define [[relations#Equivalence Relations|equivalence relation]] $\sim$ as $$x \sim y \Leftrightarrow f(x) = f(y)$$
|
||||
Relation $\sim$ is called the **(equivalence) kernel** of $f$. The [[relations#Partitions|partition]] induced by $\sim$ on $A$ is called the **coimage** of $f$ (denoted $\mathop{\text{coim}}f$). The **fiber** of an element $y$ under $F$ is $F^{-1}[\![\{y\}]\!]$, i.e. the preimage of singleton set $\{y\}$. Therefore the equivalence classes of $\sim$ are also known as the fibers of $f$.
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What kind of mathematical object is the kernel of $F \colon A \rightarrow B$?
|
||||
Back: An equivalence relation.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015583-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is the kernel of $F \colon A \rightarrow B$ defined?
|
||||
Back: As equivalence relation $\sim$ such that $x \sim y \Leftrightarrow F(x) = F(y)$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015586-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$. What name does the following relation $\sim$ go by? $$x \sim y \Leftrightarrow F(x) = F(y)$$
|
||||
Back: The kernel of $F$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015590-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$. The partition induced by the kernel of $F$ is a partition of what set?
|
||||
Back: $A$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015593-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$. What does $\mathop{\text{coim}}F$ refer to?
|
||||
Back: The coimage of $F$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015596-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is the coimage of function $F \colon A \rightarrow B$ defined?
|
||||
Back: As $A / {\sim}$ where $x \sim y \Leftrightarrow F(x) = F(y)$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015599-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$. What specific name does a member of $\mathop{\text{coim}}F$ go by?
|
||||
Back: A fiber.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015602-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$. How is the fiber of $y$ under $F$ defined?
|
||||
Back: As set $F^{-1}[\![\{y\}]\!]$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015605-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$. The fibers of $F$ make up what set?
|
||||
Back: $\mathop{\text{coim}}F$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015609-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$. How is $\mathop{\text{coim}}F$ denoted as a quotient set?
|
||||
Back: As $A / {\sim}$ where $x \sim y \Leftrightarrow F(x) = F(y)$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015613-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$ and $\sim$ be the kernel of $F$. How does $F$ factor into $\hat{F} \colon A / {\sim} \rightarrow B$?
|
||||
Back: $F = \hat{F} \circ \phi$ where $\phi$ is the natural map.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015617-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name does $\phi$ go by?
|
||||
![[function-kernel.png]]
|
||||
Back: The natural map (with respect to $\sim$).
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015620-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. How is $\phi$ defined?
|
||||
![[function-kernel.png]]
|
||||
Back: $\phi(x) = [x]_{\sim}$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015624-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name does $\sim$ go by?
|
||||
![[function-kernel.png]]
|
||||
Back: $\mathop{\text{coim}} F$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015628-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name do the members of $A / {\sim}$ go by?
|
||||
![[function-kernel.png]]
|
||||
Back: The fibers of $F$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015633-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What composition is $F$ equal to?
|
||||
![[function-kernel.png]]
|
||||
Back: $F = \hat{F} \circ \phi$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015638-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ injective?
|
||||
![[function-kernel.png]]
|
||||
Back: Yes.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015642-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ surjective?
|
||||
![[function-kernel.png]]
|
||||
Back: Not necessarily.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015646-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ bijective?
|
||||
![[function-kernel.png]]
|
||||
Back: Not necessarily.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015650-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
* “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
||||
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
* “Fiber (Mathematics),” in _Wikipedia_, April 10, 2024, [https://en.wikipedia.org/w/index.php?title=Fiber_(mathematics)&oldid=1218193490](https://en.wikipedia.org/w/index.php?title=Fiber_(mathematics)&oldid=1218193490).
|
||||
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
* “Kernel (Set Theory),” in _Wikipedia_, May 22, 2024, [https://en.wikipedia.org/w/index.php?title=Kernel_(set_theory)&oldid=1225189560](https://en.wikipedia.org/w/index.php?title=Kernel_(set_theory)&oldid=1225189560).
|
Binary file not shown.
After Width: | Height: | Size: 13 KiB |
|
@ -803,6 +803,331 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
|
|||
<!--ID: 1720969371869-->
|
||||
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).
|
||||
<!--ID: 1721098094107-->
|
||||
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).
|
||||
<!--ID: 1721223015574-->
|
||||
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).
|
||||
<!--ID: 1721098094110-->
|
||||
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).
|
||||
<!--ID: 1721098094114-->
|
||||
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).
|
||||
<!--ID: 1721098094120-->
|
||||
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).
|
||||
<!--ID: 1721098094128-->
|
||||
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).
|
||||
<!--ID: 1721098094137-->
|
||||
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).
|
||||
<!--ID: 1721098094144-->
|
||||
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).
|
||||
<!--ID: 1721098094149-->
|
||||
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).
|
||||
<!--ID: 1721098094154-->
|
||||
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).
|
||||
<!--ID: 1721098094158-->
|
||||
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).
|
||||
<!--ID: 1721098094026-->
|
||||
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).
|
||||
<!--ID: 1721098094053-->
|
||||
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).
|
||||
<!--ID: 1721098094059-->
|
||||
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).
|
||||
<!--ID: 1721098094065-->
|
||||
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).
|
||||
<!--ID: 1721098094072-->
|
||||
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).
|
||||
<!--ID: 1721098094077-->
|
||||
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).
|
||||
<!--ID: 1721098094082-->
|
||||
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).
|
||||
<!--ID: 1721098094086-->
|
||||
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).
|
||||
<!--ID: 1721098094091-->
|
||||
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).
|
||||
<!--ID: 1721098094095-->
|
||||
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).
|
||||
<!--ID: 1721098094099-->
|
||||
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).
|
||||
<!--ID: 1721098094103-->
|
||||
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)$$
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $\Pi$ be a partition of $A$. What equivalence relation $R$ is induced?
|
||||
Back: $R$ such that $xRy \Leftrightarrow (\exists B \in \Pi, x \in B \land y \in B)$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721136390215-->
|
||||
END%%
|
||||
|
||||
## Quotient Sets
|
||||
|
||||
If $R$ is an equivalence relation on $A$, then the **quotient set** "$A$ modulo $R$" is defined as $$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).
|
||||
<!--ID: 1721136390208-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Members of $A / R$ are called what?
|
||||
Back: Equivalence classes.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721218408454-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
$A / R$ is a partition of what set?
|
||||
Back: $A$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721218408484-->
|
||||
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).
|
||||
<!--ID: 1721218408508-->
|
||||
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).
|
||||
<!--ID: 1721218408514-->
|
||||
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).
|
||||
<!--ID: 1721218408520-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is quotient set $A / R$ defined?
|
||||
Back: As set $\{[x]_R \mid x \in A\}$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721218408525-->
|
||||
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).
|
||||
<!--ID: 1721218408490-->
|
||||
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).
|
||||
<!--ID: 1721218408495-->
|
||||
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).
|
||||
<!--ID: 1721218408501-->
|
||||
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).
|
||||
<!--ID: 1721218408531-->
|
||||
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).
|
||||
<!--ID: 1721218408537-->
|
||||
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).
|
||||
<!--ID: 1721218465987-->
|
||||
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).
|
||||
<!--ID: 1721219061765-->
|
||||
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).
|
||||
<!--ID: 1721223015580-->
|
||||
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).
|
||||
|
|
Loading…
Reference in New Issue