LCRS trees and set theory classes.

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@@ -127,7 +127,9 @@
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notes/_journal/2024-05-17.md

@@ -9,3 +9,5 @@ title: "2024-05-17"
 - [ ] Korean (Read 1 Story)
 
 * Exploration of the law of [[pred-trans#Distributivity of Conjunction|Distributivity of Conjunction]].
+* Flashcards for left-child, right-sibling tree representations.
+* Distinguish [[set/index#Classes|classes]] and sets. Discuss Zermelo-Fraenkel and von Neumann-Bernays alternatives.

notes/logic/prop-logic.md

@@ -238,6 +238,7 @@ Basic
 Given propositions $p$ and $q$, $p \Leftrightarrow q$ is equivalent to the conjunction of what two expressions?
 Back: $p \Rightarrow q$ and $q \Rightarrow p$.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047070-->
 END%%
 
 ## Sets
@@ -393,18 +394,21 @@ Basic
 Given sets $a$ and $b$, $a = b$ is equivalent to the conjunction of what two expressions?
 Back: $a \subseteq b$ and $b \subseteq a$.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047071-->
 END%%
 
 %%ANKI
 Cloze
 {$a \Rightarrow b$} is to propositional logic as {$a \subseteq b$} is to sets.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047073-->
 END%%
 
 %%ANKI
 Cloze
 {$a \Leftrightarrow b$} is to propositional logic as {$a = b$} is to sets.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047074-->
 END%%
 
 ## Bibliography

notes/programming/pred-trans.md

@@ -306,12 +306,14 @@ Basic
 What does Distributivity of Conjunction state?
 Back: Given command $S$ and predicates $Q$ and $R$, $wp(S, Q \land R) = wp(S, Q) \land wp(S, R)$.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047060-->
 END%%
 
 %%ANKI
 Cloze
 Distributivity of Conjunction states {$wp(S, Q \land R)$} $=$ {$wp(S, Q) \land wp(S, R)$}.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047062-->
 END%%
 
 %%ANKI
@@ -319,6 +321,7 @@ Basic
 In Gries's exposition, is Distributivity of Conjunction taken as an axiom or a theorem?
 Back: An axiom.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047064-->
 END%%
 
 %%ANKI
@@ -326,6 +329,7 @@ Basic
 Does $wp(S, Q) \land wp(S, R) \Rightarrow wp(S, Q \land R)$ hold when $S$ is nondeterministic?
 Back: Yes.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047065-->
 END%%
 
 %%ANKI
@@ -333,6 +337,7 @@ Basic
 Does $wp(S, Q \land R) \Rightarrow wp(S, Q) \land wp(S, R)$ hold when $S$ is nondeterministic?
 Back: Yes.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047067-->
 END%%
 
 %%ANKI
@@ -340,6 +345,7 @@ Basic
 What does it mean for command $S$ to be nondeterministic?
 Back: Execution may not be the same even if begun in the same state.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715969047068-->
 END%%
 
 ## Bibliography

notes/set/index.md

@@ -134,6 +134,129 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1715786028667-->
 END%%
 
+## Classes
+
+The **Zermelo-Fraenkel alternative** avoids speaking of collections defined using set theoretical notation that are not sets. The **von Neumann-Bernays** alternative calls these **classes**.
+
+%%ANKI
+Basic
+In set theory, what is a class?
+Back: A collection defined using set theoretical notation that isn't a set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576758-->
+END%%
+
+%%ANKI
+Basic
+Which two alternatives are usually employed when speaking of classes?
+Back: The Zermelo-Fraenkel alternative and the von Neumann-Bernays alternative.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576761-->
+END%%
+
+%%ANKI
+Basic
+What does the Zermelo-Fraenkel alternative say about classes?
+Back: It gives it no ontological status at all.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576763-->
+END%%
+
+%%ANKI
+Basic
+What does the von Neumann-Bernays alternative say about classes?
+Back: It refers to objects defined using set theory but that aren't actually sets.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576765-->
+END%%
+
+%%ANKI
+Cloze
+The {1:Zermelo}-{2:Fraenkel} alternative is a separate approach from the {2:von Neumann}-{1:Bernays} alternative.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576766-->
+END%%
+
+%%ANKI
+Basic
+Which set theory alternative avoids the term "class"?
+Back: The Zermelo-Fraenkel alternative.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576768-->
+END%%
+
+%%ANKI
+Basic
+Which set theory alternative embraces the term "class"?
+Back: The von Neumann-Bernays alternative.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576769-->
+END%%
+
+%%ANKI
+Basic
+What kind of mathematical object is $\{x \mid x \neq x\}$?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576771-->
+END%%
+
+%%ANKI
+Basic
+What name is given to $\{x \mid x \neq x\}$?
+Back: The empty set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576772-->
+END%%
+
+%%ANKI
+Basic
+What kind of mathematical object is $\{x \mid x = x\}$?
+Back: A class.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576774-->
+END%%
+
+%%ANKI
+Basic
+What name is given to $\{x \mid x = x\}$?
+Back: The class of all sets.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576775-->
+END%%
+
+%%ANKI
+Basic
+Are sets or classes more general?
+Back: Classes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576777-->
+END%%
+
+%%ANKI
+Basic
+Is every set a class?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576779-->
+END%%
+
+%%ANKI
+Basic
+Is every class a set?
+Back: No.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576781-->
+END%%
+
+%%ANKI
+Basic
+Assuming entrance requirement $\_\_\_$, what kind of mathematical object is $\{x \mid \_\_\_\}$?
+Back: A class.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715970576782-->
+END%%
+
 ## Bibliography
 
 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

notes/set/trees.md

@@ -722,6 +722,134 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1712409466682-->
 END%%
 
+%%ANKI
+Basic
+What $O(n)$ space representation is commonly used for ordered trees with unbounded branching?
+Back: A left-child, right-sibling tree representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969047043-->
+END%%
+
+%%ANKI
+Basic
+A node of a left-child, right-sibling tree representation has what three pointers?
+Back: The parent, left child, and right sibling.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969047046-->
+END%%
+
+%%ANKI
+Basic
+What is the space usage of a left-child, right-sibling representation?
+Back: Given $n$ nodes in the tree, $O(n)$.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969047047-->
+END%%
+
+%%ANKI
+Basic
+What space may be wasted in a $k$-child representation of a $k$-ary tree?
+Back: Some children may be absent.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969047049-->
+END%%
+
+%%ANKI
+Basic
+What space advantage does a left-child, right-sibling representation have over a $k$-child representation?
+Back: Absent children are not stored in the former.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969047051-->
+END%%
+
+%%ANKI
+Basic
+How is a `struct` of a $k$-child tree representation written?
+Back:
+```c
+struct Node {
+  struct Node *parent;
+  struct Node *children[k];
+};
+```
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: c17
+<!--ID: 1715969047052-->
+END%%
+
+%%ANKI
+Basic
+What tree representation corresponds to the following `struct`?
+```c
+struct Node {
+  struct Node *parent;
+  struct Node *children[k];
+};
+```
+Back: A $k$-ary child representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: c17
+<!--ID: 1715969047054-->
+END%%
+
+%%ANKI
+Basic
+How is a `struct` of a left-child, right-sibling tree representation written?
+Back:
+```c
+struct Node {
+  struct Node *parent;
+  struct Node *left;
+  struct Node *next;
+};
+```
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: c17
+<!--ID: 1715969047056-->
+END%%
+
+%%ANKI
+Basic
+What tree representation corresponds to the following `struct`?
+```c
+struct Node {
+  struct Node *parent;
+  struct Node *left;
+  struct Node *next;
+};
+```
+Back: A left-child, right-sibling representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: c17
+<!--ID: 1715969047057-->
+END%%
+
+%%ANKI
+Basic
+What is an LCRS tree representation?
+Back: A **l**eft-**c**hild, **r**ight-**s**ibling representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969525815-->
+END%%
+
+%%ANKI
+Basic
+The following is a portion of what kind of tree representation?
+![[lcrs-nodes.png]]
+Back: A left-child, right-sibling representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969525819-->
+END%%
+
+%%ANKI
+Basic
+The following is a portion of what kind of tree representation?
+![[binary-tree-nodes.png]]
+Back: A $k$-ary (binary) child representation.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969525820-->
+END%%
+
 ### Positional Trees
 
 A **positional tree** is a rooted tree in which each child is labeled with a specific positive integer. A **$k$-ary tree** is a positional tree with at most $k$ children/labels. A binary tree is a $2$-ary tree.
@@ -1454,6 +1582,14 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1714349367662-->
 END%%
 
+%%ANKI
+Basic
+A node of a binary tree typically has what three pointers?
+Back: The parent, left child, and right child.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715969047059-->
+END%%
+
 ## Bibliography
 
 * “Binary Tree,” in _Wikipedia_, March 13, 2024, [https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).