Notes on quotient sets, function kernels, and fibers.
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"function-surjective.png",
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"function-general.png",
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"church-rosser.png",
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"infinite-cartesian-product.png"
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"infinite-cartesian-product.png",
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"function-kernel.png"
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},
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"fields_dict": {
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"Basic": [
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@ -0,0 +1,11 @@
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---
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title: "2024-07-17"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [x] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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* Notes on [[relations#Quotient Sets|quotient sets]], function [[functions#Kernels|kernels]], and fibers.
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@ -12,3 +12,4 @@ title: "2024-07-14"
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* Initial notes on [[relations#Equivalence Relations|equivalence relations]].
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* Read chapter 2 "How to Raise Money" in "Venture Deals".
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* Finished another read of "A Cardinal Worry for Permissive Metaontology".
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* Watched [Lecture 1 "Introduction to Ontology"](https://www.youtube.com/watch?v=9AsRE437e7I).
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@ -0,0 +1,11 @@
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---
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title: "2024-07-15"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [x] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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* Notes on [[relations#Partitions|partitions]] and equivalence classes.
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@ -0,0 +1,11 @@
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---
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title: "2024-07-16"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [ ] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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* Brief notes on [[nominalism]].
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@ -93,7 +93,7 @@ END%%
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%%ANKI
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Basic
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Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
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Suppose $x, y \in A$. What set, derived from $A$, is $\langle x, y \rangle$ a member of?
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Back: $\mathscr{P}\mathscr{P}A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717679397848-->
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@ -758,8 +758,8 @@ Let $A$, $B$, and $C$ be arbitrary sets. Then
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%%ANKI
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Basic
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What kind of propositional logical statement are the monotonicity properties of $\subseteq$?
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Back: An implication.
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The monotonicity properties of $\subseteq$ are what kind of propositional logical statement?
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Back: Implications.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717073536967-->
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END%%
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@ -690,7 +690,7 @@ END%%
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%%ANKI
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Basic
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What names are usually given to the existentially quantified identifers in $o(g(n))$'s definition?
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Back: $n_0$.
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Back: $n_0$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1709519002328-->
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||||
END%%
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@ -1035,7 +1035,7 @@ END%%
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%%ANKI
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Basic
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What is the symmetric property of $\Omega$-notation?
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Back: N/A
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Back: N/A.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1709752223486-->
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END%%
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@ -557,7 +557,7 @@ END%%
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%%ANKI
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Basic
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What is the precise definition of the two's-complement of a $w$-bit number?
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What is the precise definition of the two's-complement of a $w$-bit number $x$?
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Back: The complement of $x$ with respect to $2^w$.
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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).
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<!--ID: 1709060837145-->
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@ -404,7 +404,7 @@ END%%
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%%ANKI
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Basic
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Let $h$ be a division method hash function. What does $h(10)$ evaluate to?
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Back: $10 \bmod{m}$ where $m$ is the number of slots in the hash table.
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Back: To $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720889385419-->
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@ -428,6 +428,14 @@ Tags: hashing::static
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<!--ID: 1720889385429-->
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END%%
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%%ANKI
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Basic
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Why does the division method prefer a prime number of slots?
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Back: To operate as independently as possible of the input keys.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1721218408542-->
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END%%
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%%ANKI
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Basic
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Consider hash function $h(k) = k \bmod{m}$. What method was likely used to produce this?
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@ -567,7 +567,7 @@ END%%
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%%ANKI
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Basic
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What does the Church-Rosser theorem state in terms of confluence?
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What does the Church-Rosser theorem for $\triangleright_\beta$ state in terms of confluence?
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Back: $\beta$-reduction is confluent.
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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).
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<!--ID: 1719577152613-->
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@ -50,7 +50,7 @@ $$
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%%ANKI
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Basic
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What construct is used to prove every proposition can be written in DNF or CNF?
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Back: Truth tables
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Back: Truth tables.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707311868994-->
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END%%
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@ -21,7 +21,7 @@ END%%
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%%ANKI
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Basic
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Who is attributed *the* ontological question?
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Back: Quine.
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Back: Willard Van Orman Quine.
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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).
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<!--ID: 1720912259767-->
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||||
END%%
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@ -0,0 +1,47 @@
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---
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title: Nominalism
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TARGET DECK: Obsidian::H&SS
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FILE TAGS: ontology::nominalism
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tags:
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- nominalism
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- ontology
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---
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## Overview
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**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.
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%%ANKI
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Basic
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What does it mean for a person to be anti-realist about category $X$?
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Back: They do not believe entities of $X$ exist.
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Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
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<!--ID: 1721137271035-->
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END%%
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%%ANKI
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Basic
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What does it mean for a person to be realist about category $X$?
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Back: They believe entities belonging to $X$ exist.
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Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
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<!--ID: 1721137271065-->
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END%%
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%%ANKI
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Basic
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How does Effingham define nominalism?
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Back: As anti-realism towards abstracta.
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Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
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<!--ID: 1721137271072-->
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END%%
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%%ANKI
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Cloze
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Roughly speaking, {1:permissivism} is to {2:realism} whereas {2:nominalism} is to {1:anti-realism}.
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Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
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<!--ID: 1721137271080-->
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END%%
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## Bibliography
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* Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
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@ -1327,7 +1327,7 @@ END%%
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%%ANKI
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Basic
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Consider sets $A$ and $B$. How is $A \cap B$ be rewritten as a function under some image?
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Consider sets $A$ and $B$. How is $A \cap B$ rewritten as a function under some image?
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Back: $I_A[\![B]\!]$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720885546358-->
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@ -1634,7 +1634,174 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1720819771087-->
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END%%
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## Kernels
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Let $F \colon A \rightarrow B$. Define [[relations#Equivalence Relations|equivalence relation]] $\sim$ as $$x \sim y \Leftrightarrow f(x) = f(y)$$
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Relation $\sim$ is called the **(equivalence) kernel** of $f$. The [[relations#Partitions|partition]] induced by $\sim$ on $A$ is called the **coimage** of $f$ (denoted $\mathop{\text{coim}}f$). The **fiber** of an element $y$ under $F$ is $F^{-1}[\![\{y\}]\!]$, i.e. the preimage of singleton set $\{y\}$. Therefore the equivalence classes of $\sim$ are also known as the fibers of $f$.
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%%ANKI
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Basic
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What kind of mathematical object is the kernel of $F \colon A \rightarrow B$?
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Back: An equivalence relation.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1721223015583-->
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END%%
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%%ANKI
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||||
Basic
|
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How is the kernel of $F \colon A \rightarrow B$ defined?
|
||||
Back: As equivalence relation $\sim$ such that $x \sim y \Leftrightarrow F(x) = F(y)$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1721223015586-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Let $F \colon A \rightarrow B$. What name does the following relation $\sim$ go by? $$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).
|
||||
* “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).
|
||||
|
|
|
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Reference in New Issue