Propositional and predicate logic notes.

2024-05-17 10:45:50 -06:00 · 2024-05-17 10:45:50 -06:00 · 037ea0f767
parent 5eb338be7e
commit 037ea0f767
14 changed files with 742 additions and 416 deletions
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
+++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
@ -166,7 +166,7 @@
    "_journal/2024-02-02.md": "a3b222daee8a50bce4cbac699efc7180",
    "_journal/2024-02-01.md": "3aa232387d2dc662384976fd116888eb",
    "_journal/2024-01-31.md": "7c7fbfccabc316f9e676826bf8dfe970",
-    "logic/equiv-trans.md": "63515dfd471653bc0c8bc4eca66ccaf8",
+    "logic/equiv-trans.md": "454cfcfac3a220868b3e7c4a4fd9462c",
    "_journal/2024-02-07.md": "8d81cd56a3b33883a7706d32e77b5889",
    "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
    "algorithms/loop-invariant.md": "3b390e720f3b2a98e611b49a0bb1f5a9",
@ -239,7 +239,7 @@
    "_journal/2024-02/2024-02-21.md": "f423137ae550eb958378750d1f5e98c7",
    "_journal/2024-02-23.md": "219ce9ad15a8733edd476c97628b71fd",
    "_journal/2024-02/2024-02-22.md": "312e55d57868026f6e80f7989a889c2b",
-    "c17/strings.md": "8f23005c9f8e25c72e2ac6b74b6afd7e",
+    "c17/strings.md": "192fe7fe4e321262dcd9ad2a1d322d8e",
    "c17/index.md": "78576ee41d0185df82c59999142f4edb",
    "c17/escape-sequences.md": "a8b99070336878b4e8c11e9e4525a500",
    "c17/declarations.md": "65d2faaf8cc50878ad80126109e98b1f",
@ -258,7 +258,7 @@
    "filesystems/cas.md": "d41c0d2e943adecbadd10a03fd1e4274",
    "git/objects.md": "8c1da67ac3f568624c3f9623eb2133e1",
    "git/index.md": "ca842957bda479dfa1170ae85f2f37b8",
-    "encoding/integer.md": "2ab21152b9468f547ebee496e03bc410",
+    "encoding/integer.md": "17387c74064bc738c4b4929ab37a0f56",
    "_journal/2024-02-29.md": "f610f3caed659c1de3eed5f226cab508",
    "_journal/2024-02/2024-02-28.md": "7489377c014a2ff3c535d581961b5b82",
    "_journal/2024-03-01.md": "a532486279190b0c12954966cbf8c3fe",
@ -307,7 +307,7 @@
    "_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f",
    "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b",
    "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53",
-    "set/index.md": "f2e907b2643cd4e5023169dcd96ca828",
+    "set/index.md": "fd0afffa094e47aace1ec89428218eec",
    "set/graphs.md": "4bbcea8f5711b1ae26ed0026a4a69800",
    "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb",
    "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391",
@ -395,7 +395,7 @@
    "_journal/2024-04/2024-04-28.md": "b34a9fe3bccb1f224b96ca00e78ad061",
    "programming/assertions.md": "d663ab31fc7e7296b6636edbffe9d8c7",
    "programming/text-sub.md": "4ffcaf134858b478ffc3087b58026ee8",
-    "programming/equiv-trans.md": "77b02845208902d55d3048bc05584d6e",
+    "programming/equiv-trans.md": "c165edfa15a2628d60ae3dfd839df170",
    "programming/index.md": "bb082325e269a95236aa6aff9307fe59",
    "_journal/2024-04-30.md": "369f98b9d91de89cc1f4f581bc530c0d",
    "_journal/2024-04/2024-04-29.md": "b4fa2fd62e1b4fe34c1f71dc1e9f5b0b",
@ -407,7 +407,7 @@
    "data-structures/heaps.md": "a0289574930328d422b9b53047936274",
    "data-structures/index.md": "2605977fad54956b5dc2d8dda9be2b10",
    "abstract-data-types/priority-queues.md": "d3dad736cb05c47bdc93c52a3a4af083",
-    "abstract-data-types/index.md": "6b110b20393c561497629ca4c3e09472",
+    "abstract-data-types/index.md": "ff6dbecefeec918a1def38af4999fc51",
    "_journal/2024-05-04.md": "679934dc63ded49503d573b8d2046e5d",
    "_journal/2024-05/2024-05-03.md": "be21fe4d9c1c24f74a74d06d07f8e90b",
    "_journal/2024-05/2024-05-02.md": "d7d6ba7e065d807986f0bd77281c0bb1",
@ -430,13 +430,19 @@
    "_journal/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b",
    "_journal/2024-05/2024-05-12.md": "ca9f3996272152ef89924bb328efd365",
    "git/remotes.md": "2208e34b3195b6f1ec041024a66fb38b",
-    "programming/pred-trans.md": "241dc727a7a1cb5c899e5ddc7abb92fb",
-    "set/axioms.md": "8459540f07443284f80e8fb112c69d0a",
+    "programming/pred-trans.md": "736e2c6e89494c8f5a2a7998aa905fdd",
+    "set/axioms.md": "063955bf19c703e9ad23be2aee4f1ab7",
    "_journal/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5",
    "_journal/2024-05/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b",
    "x86-64/registers.md": "be5e22ceb14084c34f446dd91702477b",
-    "_journal/2024-05-15.md": "17e60d1515be44ac4121eb86060ec4dd",
-    "_journal/2024-05/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5"
+    "_journal/2024-05-15.md": "4e6a7e6df32e93f0d8a56bc76613d908",
+    "_journal/2024-05/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5",
+    "_journal/2024-05-16.md": "580c7ec61ec56be92fa8d6affcf0a5f6",
+    "_journal/2024-05/2024-05-15.md": "4e6a7e6df32e93f0d8a56bc76613d908",
+    "logic/pred-logic.md": "c23c3da8756ac0ef17b9710a67440d84",
+    "logic/prop-logic.md": "4d45544452791470458cfb08656e16c1",
+    "_journal/2024-05-17.md": "4bada636652a610967c4ce51deb2ff43",
+    "_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c"
  },
  "fields_dict": {
    "Basic": [
--- a/notes/_journal/2024-05-17.md
+++ b/notes/_journal/2024-05-17.md
@ -0,0 +1,11 @@
+---
+title: "2024-05-17"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+
+* Exploration of the law of [[pred-trans#Distributivity of Conjunction|Distributivity of Conjunction]].
--- a/notes/_journal/2024-05/2024-05-15.md
+++ b/notes/_journal/2024-05/2024-05-15.md
@ -8,4 +8,5 @@ title: "2024-05-15"
 - [ ] Go (1 Life & Death Problem)
 - [ ] Korean (Read 1 Story)

-* Finished MOV instruction class practice problems in "Computer Systems: A Programmer's Perspective". Also notes on `leaq`.
+* Finished MOV instruction class practice problems in "Computer Systems: A Programmer's Perspective". Also notes on `leaq`.
+* Notes on different set notations.
--- a/notes/_journal/2024-05/2024-05-16.md
+++ b/notes/_journal/2024-05/2024-05-16.md
@ -0,0 +1,12 @@
+---
+title: "2024-05-16"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+
+* Refine the distinction between propositions/predicates and their state representations.
+* Additional notes on the first few set theory axioms.
--- a/notes/abstract-data-types/index.md
+++ b/notes/abstract-data-types/index.md
@ -13,7 +13,7 @@ An **abstract data type** (ADT) is a mathematical model for data types, defined
 %%ANKI
 Basic
 What is an ADT an acronym for?
-Back: **A**bstract **D**ata **T**type.
+Back: **A**bstract **D**ata **T**ype.
 Reference: “Abstract Data Type.” In _Wikipedia_, March 18, 2024. [https://en.wikipedia.org/w/index.php?title=Abstract_data_type&oldid=1214359576](https://en.wikipedia.org/w/index.php?title=Abstract_data_type&oldid=1214359576).
 <!--ID: 1714669011569-->
 END%%
--- a/notes/c17/strings.md
+++ b/notes/c17/strings.md
@ -232,7 +232,7 @@ END%%
 %%ANKI
 Basic
 How do the `+` and `␣` `printf` flags relate to one another?
-Back: Both prepend a character to positive signed-numeric types.
+Back: Both prepend a character to positively signed-numeric types.
 Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
 Tags: printf
 <!--ID: 1707918756865-->
--- a/notes/encoding/integer.md
+++ b/notes/encoding/integer.md
@ -1054,7 +1054,7 @@ END%%
 %%ANKI
 Basic
 How do you detect whether unsigned addition $s \coloneqq x +_w^u y$ overflowed?
-Back: Overflow occurs if and only if $s < x$.
+Back: Overflow occurs if and only if $s < x$ (or $s < y$).
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708799678765-->
 END%%
@ -1299,8 +1299,8 @@ END%%

 %%ANKI
 Basic
-Which two's-complement integer is its own additive inverse?
-Back: $TMin$
+Which two's-complement integers are their own additive inverse?
+Back: $TMin$ and $0$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1709040965815-->
 END%%
--- a/notes/programming/equiv-trans.md
+++ b/notes/programming/equiv-trans.md
@ -10,7 +10,9 @@ tags:

 ## Overview

-**Equivalence-transformation** refers to a class of calculi for [[propositional|propositional logic]] derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and equality ($=$). Gries covers two in "The Science of Programming": a system of evaluation and a formal system. The system of evaluation mirrors how a computer processes instructions, at least in an abstract sense. The formal system serves as a theoretical framework for reasoning about propositions and their transformations without resorting to "lower-level" operations like substitution.
+**Equivalence-transformation** refers to a class of calculi for [[prop-logic|propositional logic]] derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and equality ($=$).
+
+Gries covers two in "The Science of Programming": a system of evaluation and a formal system. The system of evaluation mirrors how a computer processes instructions, at least in an abstract sense. The formal system serves as a theoretical framework for reasoning about propositions and their transformations without resorting to "lower-level" operations like substitution.

 %%ANKI
 Basic
@ -20,14 +22,6 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1706994861286-->
 END%%

-%%ANKI
-Basic
-What are constant propositions?
-Back: Propositions that contain only constants as operands.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1707422675517-->
-END%%
-
 %%ANKI
 Cloze
 Gries replaces logical operator {$\Leftrightarrow$} in favor of {$=$}.
@ -35,27 +29,6 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1706994861295-->
 END%%

-%%ANKI
-Basic
-How does Lean define propositional equality?
-Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
-Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
-Tags: lean
-<!--ID: 1706994861298-->
-END%%
-
-%%ANKI
-Basic
-How does Lean define `propext`?
-Back:
-```lean
-axiom propext {a b : Prop} : (a ↔ b) → (a = b)
-```
-Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
-Tags: lean
-<!--ID: 1706994861300-->
-END%%
-
 %%ANKI
 Basic
 What Lean theorem justifies Gries' choice of $=$ over $\Leftrightarrow$?
@ -65,142 +38,6 @@ Tags: lean
 <!--ID: 1706994861302-->
 END%%

-%%ANKI
-Basic
-Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
-Back: Yes.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861318-->
-END%%
-
-%%ANKI
-Basic
-Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
-Back: No.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861320-->
-END%%
-
-%%ANKI
-Basic
-What proposition represents states $\{(b, T)\}$ and $\{(c, F)\}$?
-Back: $b \lor \neg c$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861337-->
-END%%
-
-%%ANKI
-Basic
-What set of states does $a \land b$ represent?
-Back: The set containing just state $\{(a, T), (b, T)\}$.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861339-->
-END%%
-
-%%ANKI
-Basic
-What is sloppy about phrase "the states in $b \lor \neg c$"?
-Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861341-->
-END%%
-
-%%ANKI
-Basic
-What is the weakest proposition?
-Back: $T$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861348-->
-END%%
-
-%%ANKI
-Basic
-What set of states does $T$ represent?
-Back: The set of all states.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861350-->
-END%%
-
-%%ANKI
-Basic
-What is the strongest proposition?
-Back: $F$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861352-->
-END%%
-
-%%ANKI
-Basic
-What set of states does $F$ represent?
-Back: The set of no states.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861354-->
-END%%
-
-%%ANKI
-Basic
-What does a proposition *represent*?
-Back: The set of states in which it is true.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861335-->
-END%%
-
-%%ANKI
-Basic
-When is $p$ stronger than $q$?
-Back: When $p \Rightarrow q$.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861343-->
-END%%
-
-%%ANKI
-Basic
-If $p \Rightarrow q$, which of $p$ or $q$ is considered stronger?
-Back: $p$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1715631869202-->
-END%%
-
-%%ANKI
-Basic
-When is $p$ weaker than $q$?
-Back: When $q \Rightarrow p$.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861346-->
-END%%
-
-%%ANKI
-Basic
-If $p \Rightarrow q$, which of $p$ or $q$ is considered weaker?
-Back: $q$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1715631869207-->
-END%%
-
-%%ANKI
-Basic
-A proposition is well-defined with respect to what?
-Back: A state to evaluate against.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861316-->
-END%%
-
-%%ANKI
-Basic
-Why is $b \land c$ stronger than $b \lor c$?
-Back: The former represents a subset of the states the latter represents.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861356-->
-END%%
-
-%%ANKI
-Basic
-What is a state?
-Back: A function mapping identifiers to values.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861314-->
-END%%
-
 %%ANKI
 Basic
 What are the two calculi Gries describes equivalence-transformation with?
--- a/notes/logic/pred-logic.md
+++ b/notes/logic/pred-logic.md
@ -0,0 +1,92 @@
+---
+title: Predicate Logic
+TARGET DECK: Obsidian::STEM
+FILE TAGS: logic::predicate
+tags:
+  - logic
+  - predicate
+---
+
+## Overview
+
+A branch of logic that uses quantified variables over non-logical objects. A **predicate** is a sentence with some number of free variables. A predicate with free variables "plugged in" is a [[prop-logic|proposition]].
+
+%%ANKI
+Cloze
+{Predicate} logic is also known as {first}-order logic.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715897257076-->
+END%%
+
+%%ANKI
+Basic
+What is a predicate?
+Back: A sentence with some number of free variables.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715897257082-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes a predicate from a proposition?
+Back: A proposition does not contain free variables.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272110-->
+END%%
+
+%%ANKI
+Basic
+How are propositions defined in terms of predicates?
+Back: A proposition is a predicate with $0$ free variables.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272115-->
+END%%
+
+%%ANKI
+Basic
+Why is "$3 + x = 12$" *not* a proposition?
+Back: Because $x$ is a variable.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272121-->
+END%%
+
+## Sets
+
+A **state** is a function that maps identifiers to values. A predicate can be equivalently seen as a representation of the set of states in which it is true.
+
+%%ANKI
+Basic
+Is $(i \geq 0)$ well-defined in $\{(i, -10)\}$?
+Back: Yes.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715898219881-->
+END%%
+
+%%ANKI
+Basic
+Is $(i \geq 0)$ well-defined in $\{(j, -10)\}$?
+Back: No.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715898219890-->
+END%%
+
+%%ANKI
+Basic
+What predicate represents states $\{(i, 0), (i, 2), (i, 4), \ldots\}$?
+Back: $i \geq 0$ is even.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715898219895-->
+END%%
+
+%%ANKI
+Basic
+What is sloppy about phrase "the states in $i + j = 0$"?
+Back: $i + j = 0$ is not a set but a representation of a set (of states).
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715898219903-->
+END%%
+
+## Bibliography
+
+* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
--- a/notes/logic/prop-logic.md
+++ b/notes/logic/prop-logic.md
@ -0,0 +1,413 @@
+---
+title: Propositional Logic
+TARGET DECK: Obsidian::STEM
+FILE TAGS: logic::propositional
+tags:
+  - logic
+  - propositional
+---
+
+## Overview
+
+A branch of logic derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and biconditionals ($\Leftrightarrow$). A **proposition** is a sentence that can be assigned a truth or false value.
+
+%%ANKI
+Cloze
+{Propositional} logic is also known as {zeroth}-order logic.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715897257085-->
+END%%
+
+%%ANKI
+Basic
+What are the basic propositional logical operators?
+Back: $\neg$, $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861291-->
+END%%
+
+%%ANKI
+Basic
+What is a proposition?
+Back: A declarative sentence which is either true or false.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272076-->
+END%%
+
+%%ANKI
+Basic
+What two categories do propositions fall within?
+Back: Atomic and molecular propositions.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272083-->
+END%%
+
+%%ANKI
+Basic
+What is an atomic proposition?
+Back: One that cannot be broken up into smaller propositions.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272087-->
+END%%
+
+%%ANKI
+Basic
+What is a molecular proposition?
+Back: One that can be broken up into smaller propositions.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272091-->
+END%%
+
+%%ANKI
+Cloze
+A {molecular} proposition can be broken up into {atomic} propositions.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272095-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes a sentence from a proposition?
+Back: The latter has an associated truth value.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272099-->
+END%%
+
+%%ANKI
+Basic
+What are constant propositions?
+Back: Propositions that contain only constants as operands.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707422675517-->
+END%%
+
+%%ANKI
+Basic
+How does Lean define propositional equality?
+Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
+Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
+Tags: lean
+<!--ID: 1706994861298-->
+END%%
+
+%%ANKI
+Basic
+How does Lean define `propext`?
+Back:
+```lean
+axiom propext {a b : Prop} : (a ↔ b) → (a = b)
+```
+Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
+Tags: lean
+<!--ID: 1706994861300-->
+END%%
+
+## Implication
+
+Implication is denoted as $\Rightarrow$. It has truth table
+
+$p$ | $q$ | $p \Rightarrow q$
+--- | --- | -----------------
+$T$ | $T$ | $T$
+$T$ | $F$ | $F$
+$F$ | $T$ | $T$
+$F$ | $F$ | $T$
+
+Implication has a few "equivalent" English expressions that are commonly used.
+Given propositions $P$ and $Q$, we have the following equivalences:
+
+* $P$ if $Q$
+* $P$ only if $Q$
+* $P$ is necessary for $Q$
+* $P$ is sufficient for $Q$
+
+%%ANKI
+Basic
+What name is given to operand $a$ in $a \Rightarrow b$?
+Back: The antecedent
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861308-->
+END%%
+
+%%ANKI
+Basic
+What name is given to operand $b$ in $a \Rightarrow b$?
+Back: The consequent
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861310-->
+END%%
+
+%%ANKI
+Basic
+How does "$P$ if $Q$" translate with $\Rightarrow$?
+Back: $Q \Rightarrow P$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272127-->
+END%%
+
+%%ANKI
+Basic
+How does "$P$ only if $Q$" translate with $\Rightarrow$?
+Back: $P \Rightarrow Q$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272134-->
+END%%
+
+%%ANKI
+Basic
+How does "$P$ is necessary for $Q$" translate with $\Rightarrow$?
+Back: $Q \Rightarrow P$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272140-->
+END%%
+
+%%ANKI
+Basic
+How does "$P$ is sufficient for $Q$" translate with $\Rightarrow$?
+Back: $P \Rightarrow Q$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272145-->
+END%%
+
+%%ANKI
+Basic
+Which of *if* or *only if* map to *necessary*?
+Back: *if*
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272151-->
+END%%
+
+%%ANKI
+Basic
+Which of *if* or *only if* map to *sufficient*?
+Back: *only if*
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272157-->
+END%%
+
+%%ANKI
+Basic
+Which logical operator maps to "if and only if"?
+Back: $\Leftrightarrow$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272163-->
+END%%
+
+%%ANKI
+Basic
+Which logical operator maps to "necessary and sufficient"?
+Back: $\Leftrightarrow$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272168-->
+END%%
+
+%%ANKI
+Basic
+What is the converse of $P \Rightarrow Q$?
+Back: $Q \Rightarrow P$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272173-->
+END%%
+
+%%ANKI
+Basic
+When is implication equivalent to its converse?
+Back: It's indeterminate.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272178-->
+END%%
+
+%%ANKI
+Basic
+What is the contrapositive of $P \Rightarrow Q$?
+Back: $\neg Q \Rightarrow \neg P$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272184-->
+END%%
+
+%%ANKI
+Basic
+When is implication equivalent to its contrapositive?
+Back: They are always equivalent.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708199272189-->
+END%%
+
+%%ANKI
+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.
+END%%
+
+## Sets
+
+A **state** is a function that maps identifiers to $T$ or $F$. A proposition can be equivalently seen as a representation of the set of states in which it is true.
+
+%%ANKI
+Basic
+What is a state?
+Back: A function mapping identifiers to values.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861314-->
+END%%
+
+%%ANKI
+Basic
+Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
+Back: Yes.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861318-->
+END%%
+
+%%ANKI
+Basic
+Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
+Back: No.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861320-->
+END%%
+
+%%ANKI
+Basic
+A proposition is well-defined with respect to what?
+Back: A state to evaluate against.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861316-->
+END%%
+
+%%ANKI
+Basic
+What proposition represents states $\{(b, T), (c, T)\}$ and $\{(b, F), (c, F)\}$?
+Back: $(b \land c) \lor (\neg b \land \neg c)$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861337-->
+END%%
+
+%%ANKI
+Basic
+What set of states does proposition $a \land b$ represent?
+Back: $\{\{(a, T), (b, T)\}\}$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861339-->
+END%%
+
+%%ANKI
+Basic
+What set of states does proposition $a \lor b$ represent?
+Back: $\{\{(a, T), (b, T)\}, \{(a, T), (b, F)\}, \{(a, F), (b, T)\}\}$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715895996324-->
+END%%
+
+%%ANKI
+Basic
+What is sloppy about phrase "the states in $b \lor \neg c$"?
+Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861341-->
+END%%
+
+%%ANKI
+Basic
+What is the weakest proposition?
+Back: $T$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861348-->
+END%%
+
+%%ANKI
+Basic
+What set of states does $T$ represent?
+Back: The set of all states.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861350-->
+END%%
+
+%%ANKI
+Basic
+What is the strongest proposition?
+Back: $F$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861352-->
+END%%
+
+%%ANKI
+Basic
+What set of states does $F$ represent?
+Back: The set of no states.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861354-->
+END%%
+
+%%ANKI
+Basic
+What does a proposition *represent*?
+Back: The set of states in which it is true.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861335-->
+END%%
+
+%%ANKI
+Basic
+When is $p$ stronger than $q$?
+Back: When $p \Rightarrow q$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861343-->
+END%%
+
+%%ANKI
+Basic
+If $p \Rightarrow q$, which of $p$ or $q$ is considered stronger?
+Back: $p$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715631869202-->
+END%%
+
+%%ANKI
+Basic
+When is $p$ weaker than $q$?
+Back: When $q \Rightarrow p$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861346-->
+END%%
+
+%%ANKI
+Basic
+If $p \Rightarrow q$, which of $p$ or $q$ is considered weaker?
+Back: $q$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715631869207-->
+END%%
+
+%%ANKI
+Basic
+Why is $b \land c$ stronger than $b \lor c$?
+Back: The former represents a subset of the states the latter represents.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861356-->
+END%%
+
+%%ANKI
+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.
+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.
+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.
+END%%
+
+## Bibliography
+
+* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
--- a/notes/logic/propositional.md
+++ b/notes/logic/propositional.md
@ -1,232 +0,0 @@
---
-title: Propositional Logic
-TARGET DECK: Obsidian::STEM
-FILE TAGS: logic::propositional
-tags:
-  - logic
-  - propositional
---
-
-## Overview
-
-A branch of logic derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and biconditionals ($\Leftrightarrow$). There exists a hierarchy of terms used to describe a string of English:
-
-* A **sentence** is any grammatical string of words.
-* A **predicate** is a sentence with free variables.
-* A **statement** is a sentence that can be assigned a truth or false value.
-	* A predicate with free variables "plugged in" is a statement.
-
-%%ANKI
-Basic
-What are the basic propositional logical operators?
-Back: $\neg$, $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861291-->
-END%%
-
-%%ANKI
-Basic
-What is a propositional statement?
-Back: A declarative sentence which is either true or false.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272076-->
-END%%
-
-%%ANKI
-Basic
-What two categories do propositional statements fall within?
-Back: Atomic and molecular statements.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272083-->
-END%%
-
-%%ANKI
-Basic
-What is an atomic statement?
-Back: One that cannot be broken up into smaller statements.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272087-->
-END%%
-
-%%ANKI
-Basic
-What is a molecular statement?
-Back: One that can be broken up into smaller statements.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272091-->
-END%%
-
-%%ANKI
-Cloze
-A {molecular} statement can be broken up into {atomic} statements.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272095-->
-END%%
-
-%%ANKI
-Basic
-What distinguishes a sentence from a statement?
-Back: The latter is a sentence that can be derived a truth value.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272099-->
-END%%
-
-%%ANKI
-Basic
-What distinguishes a predicate from a statement?
-Back: A statement does not contain free variables.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272110-->
-END%%
-
-%%ANKI
-Basic
-How are statements defined in terms of predicates?
-Back: A statement is a predicate with $0$ free variables.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272115-->
-END%%
-
-%%ANKI
-Basic
-Why is "$3 + x = 12$" *not* a statement?
-Back: Because $x$ is a variable.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272121-->
-END%%
-
-## Implication
-
-Implication is denoted as $\Rightarrow$. It has truth table
-
-$p$ | $q$ | $p \Rightarrow q$
--- | --- | -----------------
-$T$ | $T$ | $T$
-$T$ | $F$ | $F$
-$F$ | $T$ | $T$
-$F$ | $F$ | $T$
-
-Implication has a few "equivalent" English expressions that are commonly used.
-Given propositions $P$ and $Q$, we have the following equivalences:
-
-* $P$ if $Q$
-* $P$ only if $Q$
-* $P$ is necessary for $Q$
-* $P$ is sufficient for $Q$
-
-%%ANKI
-Basic
-What name is given to operand $a$ in $a \Rightarrow b$?
-Back: The antecedent
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861308-->
-END%%
-
-%%ANKI
-Basic
-What name is given to operand $b$ in $a \Rightarrow b$?
-Back: The consequent
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861310-->
-END%%
-
-%%ANKI
-Basic
-How does "$P$ if $Q$" translate with $\Rightarrow$?
-Back: $Q \Rightarrow P$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272127-->
-END%%
-
-%%ANKI
-Basic
-How does "$P$ only if $Q$" translate with $\Rightarrow$?
-Back: $P \Rightarrow Q$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272134-->
-END%%
-
-%%ANKI
-Basic
-How does "$P$ is necessary for $Q$" translate with $\Rightarrow$?
-Back: $Q \Rightarrow P$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272140-->
-END%%
-
-%%ANKI
-Basic
-How does "$P$ is sufficient for $Q$" translate with $\Rightarrow$?
-Back: $P \Rightarrow Q$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272145-->
-END%%
-
-%%ANKI
-Basic
-Which of *if* or *only if* map to *necessary*?
-Back: *if*
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272151-->
-END%%
-
-%%ANKI
-Basic
-Which of *if* or *only if* map to *sufficient*?
-Back: *only if*
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272157-->
-END%%
-
-%%ANKI
-Basic
-Which logical operator maps to "if and only if"?
-Back: $\Leftrightarrow$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272163-->
-END%%
-
-%%ANKI
-Basic
-Which logical operator maps to "necessary and sufficient"?
-Back: $\Leftrightarrow$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272168-->
-END%%
-
-%%ANKI
-Basic
-What is the converse of $P \Rightarrow Q$?
-Back: $Q \Rightarrow P$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272173-->
-END%%
-
-%%ANKI
-Basic
-When is implication equivalent to its converse?
-Back: It's indeterminate.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272178-->
-END%%
-
-%%ANKI
-Basic
-What is the contrapositive of $P \Rightarrow Q$?
-Back: $\neg Q \Rightarrow \neg P$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272184-->
-END%%
-
-%%ANKI
-Basic
-When is implication equivalent to its contrapositive?
-Back: They are always equivalent.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1708199272189-->
-END%%
-
-## Bibliography
-
-* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
--- a/notes/programming/pred-trans.md
+++ b/notes/programming/pred-trans.md
@ -200,7 +200,7 @@ END%%
 %%ANKI
 Basic
 Given command $S$ and predicate $R$, what kind of mathematical object is $wp(S, R)$?
-Back: A set (of states).
+Back: A predicate, i.e. a set of states.
 Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1715631869165-->
 END%%
@ -253,6 +253,95 @@ Reference: Reference: Gries, David. *The Science of Programming*. Texts and Mon
 <!--ID: 1715631869196-->
 END%%

+### Law of the Excluded Miracle
+
+Given any command $S$, $$wp(S, F) = F$$
+
+%%ANKI
+Basic
+Given command $S$, what does $wp(S, F)$ evaluate to?
+Back: The empty set.
+Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715806256907-->
+END%%
+
+%%ANKI
+Basic
+What does the Law of the Excluded Miracle state?
+Back: For any command $S$, $wp(S, F) = F$.
+Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715806256912-->
+END%%
+
+%%ANKI
+Basic
+What name is given to identity $wp(S, F) = F$?
+Back: The Law of the Excluded Miracle.
+Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715806256915-->
+END%%
+
+%%ANKI
+Basic
+Explain why the Law of the Excluded Miracle holds true.
+Back: No state satisfies $F$ so no precondition can either.
+Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715806256918-->
+END%%
+
+%%ANKI
+Basic
+Why is the Law of the Excluded Miracle named the way it is?
+Back: It would indeed be a miracle if execution could terminate in no state.
+Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1715806256921-->
+END%%
+
+### Distributivity of Conjunction
+
+Given command $S$ and predicates $Q$ and $R$, $$wp(S, Q \land R) = wp(S, Q) \land wp(S, R)$$
+
+%%ANKI
+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.
+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.
+END%%
+
+%%ANKI
+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.
+END%%
+
+%%ANKI
+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.
+END%%
+
+%%ANKI
+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.
+END%%
+
+%%ANKI
+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.
+END%%
+
 ## Bibliography

 * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
--- a/notes/set/axioms.md
+++ b/notes/set/axioms.md
@ -97,6 +97,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1715688034312-->
 END%%

+%%ANKI
+Basic
+How is the empty set defined using set-builder notation?
+Back: $\{x \mid x \neq x\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715900348141-->
+END%%
+
 ## Pairing Axiom

 For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
@ -165,6 +173,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1715688034329-->
 END%%

+%%ANKI
+Basic
+How is the pair set $\{u, v\}$ defined using set-builder notation?
+Back: $\{x \mid x = u \lor x = v\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715900348148-->
+END%%
+
 ## Union Axiom

 ### Preliminary Form
@ -205,12 +221,20 @@ END%%

 %%ANKI
 Basic
-What two set theory axioms proves existence of e.g. $\{x_1, x_2, x_3\}$?
+What two set theory axioms prove existence of e.g. $\{x_1, x_2, x_3\}$?
 Back: The pairing axiom and union axiom.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1715688034351-->
 END%%

+%%ANKI
+Basic
+How is the union of set $a$ and $b$ defined using set-builder notation?
+Back: $\{x \mid x \in a \lor x \in b\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715900348153-->
+END%%
+
 ## Power Set Axiom

 For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
@ -255,6 +279,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1715688034381-->
 END%%

+%%ANKI
+Basic
+How is the power set of set $a$ defined using set-builder notation?
+Back: $\{x \mid x \subseteq a\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715900348160-->
+END%%
+
 ## Bibliography

 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
--- a/notes/set/index.md
+++ b/notes/set/index.md
@ -70,6 +70,71 @@ Tags: adt::dynamic_set
 <!--ID: 1715432070083-->
 END%%

+%%ANKI
+Basic
+Define the set of prime numbers less than $10$ using abstraction.
+Back: $\{x \mid x < 10 \land x \text{ is prime}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715786028616-->
+END%%
+
+%%ANKI
+Basic
+Define the set of prime numbers less than $5$ using set-builder notation.
+Back: $\{x \mid x < 5 \land x \text{ is prime}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715786028645-->
+END%%
+
+%%ANKI
+Basic
+Define the set of prime numbers less than $5$ using roster notation.
+Back: $\{2, 3\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715786028649-->
+END%%
+
+%%ANKI
+Basic
+Define the set of prime numbers less than $5$ using abstraction.
+Back: $\{x \mid x < 5 \land x \text{ is prime}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715786028652-->
+END%%
+
+%%ANKI
+Basic
+What term describes the expression to the right of $\mid$ in set-builder notation?
+Back: The entrance requirement.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715786028656-->
+END%%
+
+%%ANKI
+Basic
+What term refers to $\_\_\; x\; \_\_$ in $\{x \mid \_\_\; x\; \_\_\}$?
+Back: The entrance requirement.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715786028659-->
+END%%
+
+%%ANKI
+Basic
+The term "entrance requirement" refers to what kind of set notation?
+Back: Set-builder/abstraction.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715786028663-->
+END%%
+
+%%ANKI
+Basic
+What name is given to set notation in which members are explicitly listed?
+Back: Roster notation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1715786028667-->
+END%%
+
 ## Bibliography

+* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).