Image operations and the assignment command.
parent
b5cac32b5b
commit
dc1ee27c43
|
|
@ -170,7 +170,7 @@
|
|||
"posix/index.md": "97b1b8ecb9a953e855a9acf0ab25b8c8",
|
||||
"posix/signals.md": "f4132369878c683bfac4d7fd863d19ba",
|
||||
"templates/daily.md": "7866014e730e85683155207a02e367d8",
|
||||
"posix/regexp.md": "887711b8fc64ca258d9987133ecf056c",
|
||||
"posix/regexp.md": "d7d1b8cde49c405d44c7e1d9343769c0",
|
||||
"journal/2024-02-04.md": "e2b5678fc53d7284b71ed6820c02b954",
|
||||
"gawk/regexp.md": "d9229f1eabe1b99e965eecaa03bee86c",
|
||||
"_templates/daily.md": "36d0b695e8a955ff1d8e980154469e03",
|
||||
|
|
@ -194,7 +194,7 @@
|
|||
"binary/index.md": "9089c6f0e86a0727cd03984f51350de0",
|
||||
"_journal/2024-02-09.md": "a798d35f0b2bd1da130f7ac766166109",
|
||||
"c/types.md": "cf3e66e5aee58a94db3fdf0783908555",
|
||||
"logic/quantification.md": "1bc37ddebe69b4a291753a6e454d81ef",
|
||||
"logic/quantification.md": "313cb37b33dfe7604ba4b8d1db4fb90f",
|
||||
"c/declarations.md": "2de27f565d1020819008ae80593af435",
|
||||
"algorithms/sorting/bubble-sort.md": "872fb23e41fb3ac36e8c46240e9a027f",
|
||||
"_journal/2024-02-10.md": "562b01f60ea36a3c78181e39b1c02b9f",
|
||||
|
|
@ -322,7 +322,7 @@
|
|||
"_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f",
|
||||
"_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b",
|
||||
"set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53",
|
||||
"set/index.md": "9a270d08a96d0c8094ffb58671e50a12",
|
||||
"set/index.md": "43b219df1822f002fdac63aa6d1c8f9a",
|
||||
"set/graphs.md": "55298be7241906cb6b61673cf0a2e709",
|
||||
"_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb",
|
||||
"_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391",
|
||||
|
|
@ -445,7 +445,7 @@
|
|||
"_journal/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b",
|
||||
"_journal/2024-05/2024-05-12.md": "ca9f3996272152ef89924bb328efd365",
|
||||
"git/remotes.md": "cbe2cd867f675f156e7fe71ec615890d",
|
||||
"programming/pred-trans.md": "bea38879a7c500bc06e6319207f2c3d4",
|
||||
"programming/pred-trans.md": "3057645553f2a762400c2929cfe926b0",
|
||||
"set/axioms.md": "063955bf19c703e9ad23be2aee4f1ab7",
|
||||
"_journal/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5",
|
||||
"_journal/2024-05/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b",
|
||||
|
|
@ -508,7 +508,7 @@
|
|||
"_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": "a57cf42b30db6c6b430b40c6d37a2af6",
|
||||
"set/relations.md": "63cdf623940e4076d127440574f6b6a2",
|
||||
"_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2",
|
||||
"_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381",
|
||||
"_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8",
|
||||
|
|
@ -534,7 +534,7 @@
|
|||
"_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": "9647f452fe31324cbb98d9ff27fbf4ba",
|
||||
"set/functions.md": "1a09c0a0f505c5f551a04f0595971d56",
|
||||
"_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4",
|
||||
"_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307",
|
||||
"lambda-calculus/beta-reduction.md": "5532f9beec9d265724a8d205326bcf67",
|
||||
|
|
@ -583,7 +583,9 @@
|
|||
"_journal/2024-07/2024-07-01.md": "7cffc27813fe7a7338e411d054ac3bd5",
|
||||
"set/bags.md": "ba7990801734f411838d7b33e7ec0542",
|
||||
"_journal/2024-07-07.md": "9ee2d5007c34cc7ff681f3d9e998eca4",
|
||||
"_journal/2024-07/2024-07-06.md": "2b794e424985f0e7d4d899163ce5733c"
|
||||
"_journal/2024-07/2024-07-06.md": "2b794e424985f0e7d4d899163ce5733c",
|
||||
"_journal/2024-07-08.md": "03ed5604e680ac9742ee99ae4b1eee8b",
|
||||
"_journal/2024-07/2024-07-07.md": "9ee2d5007c34cc7ff681f3d9e998eca4"
|
||||
},
|
||||
"fields_dict": {
|
||||
"Basic": [
|
||||
|
|
|
|||
|
|
@ -0,0 +1,11 @@
|
|||
---
|
||||
title: "2024-07-08"
|
||||
---
|
||||
|
||||
- [x] Anki Flashcards
|
||||
- [x] KoL
|
||||
- [x] OGS
|
||||
- [ ] Sheet Music (10 min.)
|
||||
- [ ] Korean (Read 1 Story)
|
||||
|
||||
* Chapter 9.1 "Assignment to Simple Variables" in "The Science of Programming."
|
||||
|
|
@ -186,6 +186,22 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
|
|||
<!--ID: 1718327812365-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
When does $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$ hold true?
|
||||
Back: Always.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023292-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
When does $\forall x, \exists y, P(x, y) \Rightarrow \exists y, \forall x, P(x, y)$ hold true?
|
||||
Back: When there exists a $y$ that $P(x, y)$ holds for over all quantified $x$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023296-->
|
||||
END%%
|
||||
|
||||
## Identifiers
|
||||
|
||||
Identifiers are said to be **bound** if they are parameters to a quantifier. Identifiers that are not bound are said to be **free**. A first-order logic formula is said to be in **prenex normal form** (PNF) if written in two parts: the first consisting of quantifiers and bound variables (the **prefix**), and the second consisting of no quantifiers (the **matrix**).
|
||||
|
|
|
|||
|
|
@ -133,7 +133,7 @@ END%%
|
|||
%%ANKI
|
||||
Basic
|
||||
`^` and `$$` belong to what operator category?
|
||||
Back: Anchors
|
||||
Back: Anchors.
|
||||
Reference: “POSIX Basic Regular Expressions,” accessed February 4, 2024, [https://en.wikibooks.org/wiki/Regular_Expressions/POSIX_Basic_Regular_Expressions](https://en.wikibooks.org/wiki/Regular_Expressions/POSIX_Basic_Regular_Expressions).
|
||||
<!--ID: 1707050923643-->
|
||||
END%%
|
||||
|
|
|
|||
|
|
@ -620,6 +620,139 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
|
|||
<!--ID: 1719019485666-->
|
||||
END%%
|
||||
|
||||
### Assignment
|
||||
|
||||
The assignment command has form $x \coloneqq e$, provided the types of $x$ and $e$ are the same. This command is read as "$x$ becomes $e$" and is defined as $$wp(''x \coloneqq e'', R) = domain(e) \,\mathop{\textbf{cand}}\, R_e^x$$
|
||||
where $domain(e)$ is a predicate that describes the set of all states in which $e$ may be evaluated.
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
The assignment command has what form?
|
||||
Back: $x \coloneqq e$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926777-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is assignment "$x \coloneqq e$" pronounced?
|
||||
Back: As "$x$ becomes $e$".
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926789-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is assignment "$x \coloneqq e$" defined in terms of $wp$?
|
||||
Back: $wp(''x \coloneqq e'', R) = domain(e) \,\mathop{\textbf{cand}}\, R_e^x$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926794-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
In the $wp$ definition of "$x \coloneqq e$", what does $domain(e)$ refer to?
|
||||
Back: A predicate that holds if $e$ is well-defined.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926799-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
In the $wp$ definition of "$x \coloneqq e$", $domain(e)$ must exclude which states?
|
||||
Back: Those in which evaluation of $e$ would be undefined.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926804-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What assumption is made when defining assignment as "$wp(''x \coloneqq e'', R) = R_e^x$"?
|
||||
Back: $domain(e)$, i.e. evaluation of $e$ is well-defined.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926808-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is definition "$wp(''x \coloneqq e'', R) = R_e^x$" more completely stated?
|
||||
Back: $wp(''x \coloneqq e'', R) = domain(e) \,\mathop{\textbf{cand}}\, R_e^x$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926813-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
In what way is the $wp$ definition of assignment usually simplified?
|
||||
Back: It is assumed evaluation of expressions (i.e. the RHS of $\coloneqq$) are always well-defined.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926818-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What does Gries state is "bewildering at first" about definition "$wp(''x \coloneqq e'', R) = R_e^x$"?
|
||||
Back: Operational habits make us feel the precondition should be $R$ and postcondition $R_e^x$.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926824-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is definition $wp(''x \coloneqq e'', R) = R_e^x$ informally justified?
|
||||
Back: Since $x$ becomes $e$, $R$ is true after execution iff $R_e^x$ is true before execution.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926828-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What does $wp(''x \coloneqq 5'', x = 5)$ evaluate to?
|
||||
Back: $T$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926833-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What does $wp(''x \coloneqq 5'', x \neq 5)$ evaluate to?
|
||||
Back: $F$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926837-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What does $wp(''x \coloneqq x + 1'', x < 0)$ evaluate to?
|
||||
Back: $x < -1$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926842-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Given array $b$ with subscript range $0{:}100$, what does $wp(''x \coloneqq b[i]'', x = b[i])$ evaluate to?
|
||||
Back: $0 \leq i \leq 100$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926846-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Assume $c$ is constant and $x, y$ are distinct. What does $wp(''x \coloneqq e'', y = c)$ evaluate to?
|
||||
Back: $y = c$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926852-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What does evaluation "$wp(''x \coloneqq e'', y = c) = (y = c)$" demonstrate?
|
||||
Back: That assignment (and expression evaluation) should exhibit no side effects.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720447926858-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
|
|
@ -1266,6 +1266,142 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
|
|||
<!--ID: 1719103644321-->
|
||||
END%%
|
||||
|
||||
The following holds for any sets $F$, $A$, $B$, and $\mathscr{A}$:
|
||||
|
||||
* The image of unions is the union of the images:
|
||||
* $F[\![\bigcup\mathscr{A}]\!] = \bigcup\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$
|
||||
* The image of intersections is a subset of the intersection of images:
|
||||
* $F[\![\bigcap \mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$
|
||||
* Equality holds if $F$ is single-rooted.
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the image of unions relate to the union of images?
|
||||
Back: They are equal.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720382880557-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the union of images relate to the images of unions?
|
||||
Back: They are equal.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023254-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does $F[\![A \cup B]\!]$ relate to $F[\![A]\!] \cup F[\![B]\!]$?
|
||||
Back: They are equal.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720382880566-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What is the generalization of identity $F[\![A \cup B]\!] = F[\![A]\!] \cup F[\![B]\!]$?
|
||||
Back: $F[\![\bigcup\mathscr{A}]\!] = \bigcup\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720382880569-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What is the specialization of identity $F[\![\bigcup\mathscr{A}]\!] = \bigcup\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$?
|
||||
Back: $F[\![A \cup B]\!] = F[\![A]\!] \cup F[\![B]\!]$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720382880572-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
*Why* is the following identity intuitively true? $$F[\![A \cup B]\!] = F[\![A]\!] \cup F[\![B]\!]$$
|
||||
Back: $F(x)$ is in the range of $F$ regardless of whether $x \in A$ or $x \in B$ (or both).
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720382880575-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the image of intersections relate to the intersection of images?
|
||||
Back: The former is a subset of the latter.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023257-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the intersection of images relate to the image of intersections?
|
||||
Back: The latter is a subset of the former.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023261-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What condition on set $F$ makes the following true? $F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$
|
||||
Back: N/A. This is always true.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023264-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What condition on set $F$ makes the following true? $F[\![A \cap B]\!] = F[\![A]\!] \cap F[\![B]\!]$
|
||||
Back: $F$ is single-rooted.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023267-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What condition on set $F$ makes the following true? $F[\![A]\!] \cap F[\![B]\!] \subseteq F[\![A \cap B]\!]$
|
||||
Back: $F$ is single-rooted.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023270-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What is the generalization of identity $F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$?
|
||||
Back: $F[\![\bigcap\mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023273-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What is the specialization of identity $F[\![\bigcap\mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$?
|
||||
Back: $F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023276-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
*Why* is the following identity intuitively true? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$
|
||||
Back: $A \cap B$ could be empty but $F[\![A]\!] \cap F[\![B]\!]$ could be nonempty.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023280-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
The following is analagous to what logical expression of commuting quantifiers? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$
|
||||
Back: $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023284-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Given single-rooted $R$, the following is analagous to what logical expression of commuting quantifiers? $$R[\![A \cap B]\!] = R[\![A]\!] \cap R[\![B]\!]$$
|
||||
Back: $\exists x, \forall y, P(x, y) \Leftrightarrow \forall y, \exists x, P(x, y)$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023288-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
* “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
||||
|
|
|
|||
|
|
@ -305,6 +305,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
|
|||
<!--ID: 1720370610028-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How do you rewrite $\{A \mid A \in B\}$ with an existential in the entrance requirement?
|
||||
Back: $\{v \mid A \in B \land v = A\}$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720381621849-->
|
||||
END%%
|
||||
|
||||
## Extensionality
|
||||
|
||||
If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
|
||||
|
|
|
|||
|
|
@ -230,7 +230,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
The following is analagous to what predicate logical expression of commuting quantifiers?$$\mathop{\text{dom}}\bigcup\mathscr{A} = \bigcup\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
|
||||
The following is analagous to what logical expression of commuting quantifiers?$$\mathop{\text{dom}}\bigcup\mathscr{A} = \bigcup\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
|
||||
Back: $\exists x, \exists y, P(x, y) \Leftrightarrow \exists y, \exists x, P(x, y)$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1718327739907-->
|
||||
|
|
@ -245,7 +245,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
The following is analagous to what predicate logical expression of commuting quantifiers? $$\mathop{\text{dom}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
|
||||
The following is analagous to what logical expression of commuting quantifiers? $$\mathop{\text{dom}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{dom}} R \mid R \in \mathscr{A}\}$$
|
||||
Back: $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1718327739914-->
|
||||
|
|
@ -275,7 +275,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
The following is analagous to what predicate logical expression of commuting quantifiers? $$\mathop{\text{ran}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$$
|
||||
The following is analagous to what logical expression of commuting quantifiers? $$\mathop{\text{ran}}\bigcap\mathscr{A} \subseteq \bigcap\, \{\mathop{\text{ran}} R \mid R \in \mathscr{A}\}$$
|
||||
Back: $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1718327739931-->
|
||||
|
|
|
|||
Loading…
Reference in New Issue