Notes on binary search trees.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -146,7 +146,15 @@
     "infinite-cartesian-product.png",
     "function-kernel.png",
     "triple-table-repr.png",
-    "binary-search-tree.png"
+    "binary-search-tree.png",
+    "bst-right-child.png",
+    "bst-right-child-after.png",
+    "bst-left-child.png",
+    "bst-left-child-after.png",
+    "bst-right-succ.png",
+    "bst-right-succ-after.png",
+    "bst-deep-succ.png",
+    "bst-deep-succ-after.png"
   ],
   "File Hashes": {
     "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8",
@@ -253,7 +261,7 @@
     "combinatorics/inclusion-exclusion.md": "c27b49ee03cc5ee854d0e8bd12a1d505",
     "_journal/2024-02-21.md": "b9d944ecebe625da5dd72aeea6a916a2",
     "_journal/2024-02/2024-02-20.md": "af2ef10727726200c4defe2eafc7d841",
-    "algebra/radices.md": "662a3e2f611dd9655f1ef6c7ed0a9943",
+    "algebra/radices.md": "474178afb07f3d5037c1547cc1a132f2",
     "_journal/2024-02-22.md": "e01f1d4bd2f7ac2a667cdfd500885a2a",
     "_journal/2024-02/2024-02-21.md": "f423137ae550eb958378750d1f5e98c7",
     "_journal/2024-02-23.md": "219ce9ad15a8733edd476c97628b71fd",
@@ -464,7 +472,7 @@
     "_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c",
     "_journal/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c",
     "hashing/direct-addressing.md": "f75cc22e74ae974fe4f568a2ee9f951f",
-    "hashing/index.md": "90be101304334dc8771ca141fae0092c",
+    "hashing/index.md": "760d1fad7a890814afecfd29c6675f67",
     "set/classes.md": "6776b4dc415021e0ef60b323b5c2d436",
     "_journal/2024-05-19.md": "fddd90fae08fab9bd83b0ef5d362c93a",
     "_journal/2024-05/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c",
@@ -487,7 +495,7 @@
     "_journal/2024-05/2024-05-25.md": "3e8a0061fa58a6e5c48d12800d1ab869",
     "_journal/2024-05-27.md": "b36636d10eab34380f17f288868df3ae",
     "_journal/2024-05/2024-05-26.md": "abe84b5beae74baa25501c818e64fc95",
-    "algebra/set.md": "f7a775cd9f6bf0b3a127fc9c3b9a3c15",
+    "algebra/set.md": "b4d5ce4b914abe5063bf6043cf7a0aee",
     "algebra/boolean.md": "fc47edb7d0080b73ce1ce0d3e0e16d7d",
     "git/merge-conflicts.md": "761ad6137ec51d3877f7d5b3615ca5cb",
     "_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337",
@@ -512,7 +520,7 @@
     "_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": "7ed656318eb661b9cc65d4007311f392",
+    "set/relations.md": "84a66a74805014d3b88ea5e8a3e763bc",
     "_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2",
     "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381",
     "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8",
@@ -651,8 +659,8 @@
     "_journal/2024-07/2024-07-26.md": "c167f734a5037e1a5537b1e95ca6790f",
     "_journal/2024-07-28.md": "8a2393673132ac57a86b3b528bfc4a16",
     "_journal/2024-07/2024-07-27.md": "7c48690746d8320494e29e92390eb6ee",
-    "ontology/rdf/uri.md": "5d9f355f314a54c5fb5099d751070656",
-    "ontology/rdf/index.md": "4875a042ed4e2ff0448e03bb17f1a736",
+    "ontology/rdf/uri.md": "07999207f65ffb35f55af42c9922e7c5",
+    "ontology/rdf/index.md": "36424c9bad6088cdee67f74e3b8a019f",
     "ontology/philosophy/permissivism.md": "643e815a79bc5c050cde9f996aa44ef5",
     "ontology/philosophy/nominalism.md": "46245c644238157e15c7cb6def27d90a",
     "ontology/philosophy/index.md": "6c7c60f91f78fdc1cdd8c012b1ac4ebd",
@@ -661,11 +669,14 @@
     "_journal/2024-07/2024-07-28.md": "ff5dcfb3dc1b5592894363414e20b02f",
     "_journal/2024-07-30.md": "025194b9b770b56a81b5a52d96a305f2",
     "_journal/2024-07/2024-07-29.md": "ab496a55aacc60d9456378920c599871",
-    "data-structures/binary-search-tree.md": "c2076aa12f7afacc96fa504dd89dae5e",
+    "data-structures/binary-search-tree.md": "34127d93a49d7c947008a37fecfe74ac",
     "_journal/2024-08-01.md": "2e3da352cfbaf29b6b49e3c3a4f090df",
     "_journal/2024-07/2024-07-31.md": "d397c5a4d42660eeaa290aa8316d55c1",
     "_journal/2024-07/2024-07-30.md": "025194b9b770b56a81b5a52d96a305f2",
-    "_journal/2024-08-02.md": "24648f61c675c1c52e4cb19cbac6f0dc"
+    "_journal/2024-08-02.md": "24648f61c675c1c52e4cb19cbac6f0dc",
+    "_journal/2024-08-03.md": "27f1316f792f7673cd7ea1b1e44444be",
+    "_journal/2024-08/2024-08-02.md": "076c35545f292eddd3c7253a41fbd40c",
+    "_journal/2024-08/2024-08-01.md": "2e3da352cfbaf29b6b49e3c3a4f090df"
   },
   "fields_dict": {
     "Basic": [

notes/_journal/2024-08-03.md

@@ -2,8 +2,10 @@
 title: "2024-08-03"
 ---
 
-- [ ] Anki Flashcards
+- [x] Anki Flashcards
 - [x] KoL
 - [ ] OGS
 - [ ] Sheet Music (10 min.)
-- [ ] Korean (Read 1 Story)
+- [ ] Korean (Read 1 Story)
+
+* Additional notes on binary search trees.

notes/algebra/radices.md

@@ -277,7 +277,7 @@ END%%
 %%ANKI
 Basic
 Which hexadecimal digits are encoded in binary with a leading `1` bit?
-Back: `8` through `F`
+Back: `8` through `F`.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: binary::hex
 <!--ID: 1708631918825-->
@@ -286,7 +286,7 @@ END%%
 %%ANKI
 Basic
 Which hexadecimal digits are encoded in binary with a leading `0` bit?
-Back: `0` through `7`
+Back: `0` through `7`.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: binary::hex
 <!--ID: 1708631918829-->

notes/data-structures/binary-search-tree.md

@@ -13,6 +13,8 @@ A binary search tree (BST) is a [[trees#Binary Trees|binary tree]] satisfying th
 
 > Let $x$ be a node in a binary search tree. If $y$ is a node in the left subtree of $x$, then $y.key \leq x.key$. If $y$ is a node in the right subtree of $x$, then $y.key \geq x.key$.
 
+## Traversals
+
 Consider an arbitrary node $x$ of some BST. Then:
 
 * An **inorder** traversal visits $x$'s left child, then $x$, then $x$'s right child.
@@ -190,6 +192,398 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1722342235784-->
 END%%
 
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime of search?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303021-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for finding the minimum?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303023-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for finding the maximum?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303025-->
+END%%
+
+### Successors
+
+The **successor** of a node in a binary search tree is the node whose value would appear immediately after in an in-order traversal.
+
+%%ANKI
+Basic
+How do we define the successor of a BST node in terms of in-order traversals?
+Back: As the node encountered immediately after in an in-order traversal.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623599-->
+END%%
+
+%%ANKI
+Basic
+Which node is the successor of $7$ in the following BST?
+![[binary-search-tree.png]]
+Back: $8$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623601-->
+END%%
+
+%%ANKI
+Basic
+Which node is the successor of $2$ in the following BST?
+![[binary-search-tree.png]]
+Back: $5$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623603-->
+END%%
+
+%%ANKI
+Basic
+Which node is the successor of $5$ in the following BST?
+![[binary-search-tree.png]]
+Back: Either $5$ or $6$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623604-->
+END%%
+
+%%ANKI
+Basic
+Which node(s) in a BST have no successor?
+Back: The maximum.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623606-->
+END%%
+
+%%ANKI
+Basic
+What two cases are considered when finding the successor of a BST node?
+Back: If the node has a right subtree or not.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623607-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a right subtree. What is its successor?
+Back: The minimum of the right subtree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623608-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node does not have a right subtree. What is its successor?
+Back: The first proper ancestor reached from the LHS.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623610-->
+END%%
+
+%%ANKI
+Cloze
+If a BST node has a right subtree, it's successor cannot have a {left} subtree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303026-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a right subtree. *Why* can't its successor have a left subtree?
+Back: Because then a node in that left subtree would be the actual successor.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303028-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a right subtree. *Why* can't its successor have a right subtree?
+Back: N/A. It can.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303029-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for finding a node's successor?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303030-->
+END%%
+
+### Predecessors
+
+The **predecessor** of a node in a binary search tree is the node whose value would appear immediately before in an in-order traversal.
+
+%%ANKI
+Basic
+How do we define the predecessor of a BST node in terms of in-order traversals?
+Back: As the node encountered immediately before in an in-order traversal.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623611-->
+END%%
+
+%%ANKI
+Basic
+Which node is the predecessor of $7$ in the following BST?
+![[binary-search-tree.png]]
+Back: $6$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623613-->
+END%%
+
+%%ANKI
+Basic
+Which node is the predecessor of $2$ in the following BST?
+![[binary-search-tree.png]]
+Back: N/A.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623614-->
+END%%
+
+%%ANKI
+Basic
+Which node is the predecessor of $5$ in the following BST?
+![[binary-search-tree.png]]
+Back: Either $2$ or $5$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623615-->
+END%%
+
+%%ANKI
+Basic
+Which node(s) in a BST have no predecessor?
+Back: The minimum.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623616-->
+END%%
+
+%%ANKI
+Basic
+What two cases are considered when finding the predecessor of a BST node?
+Back: If the node has a left subtree or not.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623617-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a left subtree. What is its predecessor?
+Back: The maximum of the left subtree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623618-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node does not have a left subtree. What is its predecessor?
+Back: The first proper ancestor reached from the RHS.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623619-->
+END%%
+
+%%ANKI
+Cloze
+If a BST node has a left subtree, it's predecessor cannot have a {right} subtree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303031-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a left subtree. *Why* can't its successor have a left subtree?
+Back: N/A. It can.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303033-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a left subtree. *Why* can't its predecessor have a right subtree?
+Back: Because then a node in that right subtree would be the actual predecessor.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303034-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for finding a node's predecessor?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303035-->
+END%%
+
+## Deletions
+
+Consider deleting node $z$ from a BST. There are three conceptual cases to consider corresponding to the number of children $z$ has:
+
+* If $z$ has no children we can just replace $z$ with `NIL`.
+* If $z$ has one child, we can replace $z$ with its child.
+* If $z$ has two children, we swap $z$ with either its predecessor or successor, updating pointers as necessary to maintain the binary-search-tree property.
+
+%%ANKI
+Basic
+Insert node $7.5$ into the following BST. Where is the new node located?
+![[binary-search-tree.png]]
+Back: As $8$'s left child.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303036-->
+END%%
+
+%%ANKI
+Basic
+Insert node $6.5$ into the following BST. Where is the new node located?
+![[binary-search-tree.png]]
+Back: As $7$'s left child.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303037-->
+END%%
+
+%%ANKI
+Basic
+Insert node $5.5$ into the following BST. Where is the new node located?
+![[binary-search-tree.png]]
+Back: As the lower $5$'s right child.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303038-->
+END%%
+
+%%ANKI
+Basic
+When does insertion into a BST modify the root node?
+Back: When the tree being inserted into is empty.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303039-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for inserting a node?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303040-->
+END%%
+
+%%ANKI
+Basic
+How many cases are there to consider when deleting a node from a BST?
+Back: Three.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303041-->
+END%%
+
+%%ANKI
+Basic
+What corresponds to the cases to consider when deleting a node from a BST?
+Back: The number of children the node in question has.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303042-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with no children. Which node is $z$ replaced with?
+Back: `NIL`
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303043-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with one child. Which node is $z$ replaced with?
+Back: Its one child.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303044-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with two children. Which node is $z$ replaced with?
+Back: Either its successor or predecessor.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303045-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with two children. If replacing with its successor, what two subcases need to be considered?
+Back: If $z$'s successor is its right child or not.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303046-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with two children. If replacing with its predecessor, what two subcases need to be considered?
+Back: If $z$'s predessor is its left child or not.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303047-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for deleting a node?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303048-->
+END%%
+
+%%ANKI
+Basic
+Delete $z$ from the following BST. What does the resulting tree look like?
+![[bst-right-child.png]]
+Back:
+![[bst-right-child-after.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303049-->
+END%%
+
+%%ANKI
+Basic
+Delete $z$ from the following BST. What does the resulting tree look like?
+![[bst-left-child.png]]
+Back:
+![[bst-left-child-after.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303050-->
+END%%
+
+%%ANKI
+Basic
+Delete $z$ from the following BST. What does the resulting tree look like?
+![[bst-right-succ.png]]
+Back:
+![[bst-right-succ-after.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303051-->
+END%%
+
+%%ANKI
+Basic
+Delete $z$ from the following BST. What does the resulting tree look like?
+![[bst-deep-succ.png]]
+Back:
+![[bst-deep-succ-after.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303052-->
+END%%
+
+%%ANKI
+Basic
+What arbitrary choice was implied in the deletion algorithm of the following BST?
+![[bst-deep-succ.png]]
+Back: To replace deleted nodes with their successor instead of predecessor.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713414719-->
+END%%
+
 ## Bibliography
 
 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/hashing/index.md

@@ -328,7 +328,7 @@ END%%
 %%ANKI
 Basic
 Let $h$ be a division method hash function. What does $h(10)$ evaluate to?
-Back: To $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
+Back: $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 Tags: hashing::static
 <!--ID: 1720889385419-->

notes/ontology/rdf/index.md

@@ -123,7 +123,7 @@ END%%
 
 %%ANKI
 Basic
-RDF identifiers are represented using what?
+RDF identifiers use what encoding?
 Back: URIs.
 Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
 <!--ID: 1722191359882-->

notes/ontology/rdf/uri.md

@@ -67,7 +67,7 @@ where the `<authority>` is further composed as
 Cloze
 A URI has the following generic syntax:
 
-{`<schema>`}`:`{`[//<authority>]`}`]`{`<path>`}{`[?<query>]`}{`[#<fragment>]`}.
+{`<scheme>`}`:`{`[//<authority>]`}{`<path>`}{`[?<query>]`}{`[#<fragment>]`}.
 Reference: “Uniform Resource Identifier.” In _Wikipedia_, July 22, 2024. [https://en.wikipedia.org/w/index.php?title=Uniform_Resource_Identifier](https://en.wikipedia.org/w/index.php?title=Uniform_Resource_Identifier&oldid=1235957234).
 <!--ID: 1722211276499-->
 END%%
@@ -488,7 +488,7 @@ A **compact URI** (CURIE) is a denser representation of URIs. In its simplest fo
 %%ANKI
 Basic
 URIs in RDF are usually represented in what condensed format?
-Back: CURIE.
+Back: CURIEs.
 Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
 <!--ID: 1722191359875-->
 END%%
@@ -496,7 +496,7 @@ END%%
 %%ANKI
 Basic
 What is CURIE an acronym for?
-Back: **C**ompact **URI**&.
+Back: **C**ompact **URI**s.
 Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
 <!--ID: 1722191359876-->
 END%%

notes/set/relations.md

@@ -647,7 +647,7 @@ END%%
 %%ANKI
 Basic
 Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $a$?
-Back: N/A. We must ask if $R$ is reflexive on a set.
+Back: N/A. We should ask if $R$ is reflexive on set $\{a\}$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1720967429808-->
 END%%
@@ -729,7 +729,7 @@ END%%
 %%ANKI
 Basic
 Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ irreflexive on $a$?
-Back: N/A. We must ask if $R$ is irreflexive on a set.
+Back: N/A. We should ask if $R$ is irreflexive on set $\{a\}$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721870888391-->
 END%%