Image operations and the assignment command.

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notes/_journal/2024-07-08.md

@@ -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."

notes/logic/quantification.md

@@ -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**).

notes/posix/regexp.md

@@ -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%%

notes/programming/pred-trans.md

@@ -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.

notes/set/functions.md

@@ -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).

notes/set/index.md

@@ -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: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$

notes/set/relations.md

@@ -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-->