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Image operations and the assignment command.

c-declarations
Joshua Potter 2024-07-08 08:13:15 -06:00
parent b5cac32b5b
commit dc1ee27c43
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"Basic": [

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---
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."

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@ -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**).

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@ -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%%

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@ -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.

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@ -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).

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

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@ -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-->