Notes on proof methods.

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     "Basic": [

notes/_journal/2024-07-24.md

@@ -0,0 +1,11 @@
+---
+title: "2024-07-24"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Expand on a variety of [[proofs/index|proof methods]].

notes/_journal/2024-07/2024-07-22.md

@@ -9,4 +9,5 @@ title: "2024-07-22"
 - [ ] Korean (Read 1 Story)
 
 * Beginning notes on [[natural-deduction|natural deduction]].
-* Read chapter 3 "Overview of the Term Sheet" of "Venture Deals".
+* Read chapter 3 "Overview of the Term Sheet" of "Venture Deals".
+* Finished "Equivalence Relations" exercises of "Elements of Set Theory".

notes/_journal/2024-07/2024-07-23.md

@@ -0,0 +1,11 @@
+---
+title: "2024-07-23"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Worked through first half of Protege [New Pizza Tutorial](https://www.michaeldebellis.com/post/new-protege-pizza-tutorial).

notes/algebra/radices.md

@@ -276,7 +276,7 @@ END%%
 
 %%ANKI
 Basic
-Which hexadecimal digits encode binary with a leading `1` bit?
+Which hexadecimal digits are encoded in binary with a leading `1` bit?
 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
@@ -285,7 +285,7 @@ END%%
 
 %%ANKI
 Basic
-Which hexadecimal digits encode binary with a leading `0` bit?
+Which hexadecimal digits are encoded in binary with a leading `0` bit?
 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

notes/algebra/set.md

@@ -293,8 +293,8 @@ END%%
 
 %%ANKI
 Basic
-Under what two conditions is $A \times B = B \times A$?
+Under what three conditions is $A \times B = B \times A$?
-Back: $A = B$ or either set is the empty set.
+Back: $A = B$ or $A = \varnothing$ or $B = \varnothing$.
 Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
 <!--ID: 1718069881709-->
 END%%

notes/formal-system/logical-system/pred-logic.md

@@ -114,64 +114,52 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1707494819970-->
 END%%
 
-### Universals
+#### Uniqueness
 
-**Universal quantification** ($\forall$) asserts that every member of a set satisfies a property.
+We can also denote existence and uniqueness using $\exists!$. For example, $\exists! x, P(x)$ indicates there exists a unique $x$ satisfying $P(x)$, i.e. there is exactly one $x$ such that $P(x)$ holds: $$(\exists! x, P(x)) = (\exists x, P(x)) \land (\forall x, \forall y, (P(x) \land P(y)) \Rightarrow (x = y)))$$
+The first conjunct denotes existence while the second denotes uniqueness.
 
 %%ANKI
 Basic
-What symbol denotes universal quantification?
+What non-counting quantifer denotes unique existential quantification?
-Back: $\forall$
+Back: $\exists!$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
-<!--ID: 1707494819971-->
+<!--ID: 1721824073159-->
 END%%
 
 %%ANKI
 Basic
-How many members in the domain of discourse must satisfy a property in universal quantification?
+Unique existential quantification can be expressed using what counting quantification?
-Back: All of them.
+Back: $\exists^{=1}$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
-<!--ID: 1707494819973-->
+<!--ID: 1721824073162-->
 END%%
 
 %%ANKI
 Basic
-$\forall x : S, P(x)$ is shorthand for what?
+How is $\exists! x, P(x)$ expanded using the basic existential and universal quantifiers?
-Back: $\forall x, x \in S \Rightarrow P(x)$
+Back: $(\exists x, P(x)) \land (\forall x, \forall y, (P(x) \land P(y)) \Rightarrow (x = y))$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
-<!--ID: 1707494819976-->
+<!--ID: 1721824073165-->
 END%%
 
 %%ANKI
 Basic
-What is the identity element of $\land$?
+How do we write the equivalent existence (not uniqueness) assertion made by $\exists! x, P(x)$?
-Back: $T$
+Back: $\exists x, P(x))$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
-<!--ID: 1707494819978-->
+<!--ID: 1721824073168-->
-END%%
-
-%%ANKI
-Cloze
-{1:$\exists$} is to {2:$\lor$} as {2:$\forall$} is to {1:$\land$}.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1707494819979-->
 END%%
 
 %%ANKI
 Basic
-How is $\forall x : S, P(x)$ equivalently written in terms of existential quantification?
+How do we write the equivalent uniqueness (not existence) assertion made by $\exists! x, P(x)$?
-Back: $\neg \exists x : S, \neg P(x)$
+Back: $\forall x, \forall y, (P(x) \land P(y)) \Rightarrow (x = y)$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
-<!--ID: 1707494819981-->
+<!--ID: 1721824073172-->
 END%%
 
-%%ANKI
+#### Counting
-How is $\exists x : S, P(x)$ equivalently written in terms of universal quantification?
-Back: $\neg \forall x : S, \neg P(x)$
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-END%%
-
-### Counting
 
 **Counting quantification** ($\exists^{=k}$ or $\exists^{\geq k}$) asserts that (at least) $k$ (say) members of a set satisfy a property.
 
@@ -251,6 +239,63 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1720386023296-->
 END%%
 
+### Universals
+
+**Universal quantification** ($\forall$) asserts that every member of a set satisfies a property.
+
+%%ANKI
+Basic
+What symbol denotes universal quantification?
+Back: $\forall$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707494819971-->
+END%%
+
+%%ANKI
+Basic
+How many members in the domain of discourse must satisfy a property in universal quantification?
+Back: All of them.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707494819973-->
+END%%
+
+%%ANKI
+Basic
+$\forall x : S, P(x)$ is shorthand for what?
+Back: $\forall x, x \in S \Rightarrow P(x)$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707494819976-->
+END%%
+
+%%ANKI
+Basic
+What is the identity element of $\land$?
+Back: $T$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707494819978-->
+END%%
+
+%%ANKI
+Cloze
+{1:$\exists$} is to {2:$\lor$} as {2:$\forall$} is to {1:$\land$}.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707494819979-->
+END%%
+
+%%ANKI
+Basic
+How is $\forall x : S, P(x)$ equivalently written in terms of existential quantification?
+Back: $\neg \exists x : S, \neg P(x)$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707494819981-->
+END%%
+
+%%ANKI
+How is $\exists x : S, P(x)$ equivalently written in terms of universal quantification?
+Back: $\neg \forall x : S, \neg P(x)$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+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**).
@@ -314,4 +359,5 @@ 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).
+* 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).
+* Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

notes/formal-system/proof-system/natural-deduction.md

@@ -181,8 +181,8 @@ END%%
 
 ### Implication
 
-For propositions $E1$ and $E2$, $${\Rightarrow}{\text{-}}I: \quad \text{TODO}$$
+For propositions $E_1, \ldots, E_n$, $${\Rightarrow}{\text{-}}I: \quad \begin{array}{c} \text{from } E_1, \cdots, E_n \text{ infer } E \\ \hline (E_1 \land \cdots \land E_n) \Rightarrow E \end{array}$$
-and $${\Rightarrow}{\text{-}}E: \quad \begin{array}{c} E1 \Rightarrow E2, E1 \\ \hline E2 \end{array}$$
+and $${\Rightarrow}{\text{-}}E: \quad \begin{array}{c} E_1 \Rightarrow E_2, E_1 \\ \hline E_2 \end{array}$$
 
 %%ANKI
 Basic
@@ -192,6 +192,14 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1721665510225-->
 END%%
 
+%%ANKI
+Basic
+How is ${\Rightarrow}{\text{-}}I$ expressed in schematic notation?
+Back: $${\Rightarrow}{\text{-}}I: \quad \begin{array}{c} \text{from } E_1, \cdots, E_n \text{ infer } E \\ \hline (E_1 \land \cdots \land E_n) \Rightarrow E \end{array}$$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1721785548092-->
+END%%
+
 %%ANKI
 Basic
 In natural deduction, how is implication elimination denoted?
@@ -308,6 +316,14 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1721666244364-->
 END%%
 
+### Existential Quantification
+
+TODO
+
+### Universal Quantification
+
+TODO
+
 ## Bibliography
 
 * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

notes/hashing/index.md

@@ -596,7 +596,7 @@ Let $\mathscr{H}$ be a finite family of hash functions that map a given universe
 
 %%ANKI
 Basic
-Which of universal hashing or random hashing more general?
+Which of universal hashing or random hashing is more general?
 Back: Random hashing.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 Tags: hashing::random hashing::universal
@@ -624,7 +624,7 @@ END%%
 %%ANKI
 Basic
 Consider a hash table with $m = 1$ slot. Which hash function families are universal?
-Back: Finite families of hash functions mapping to e.g. $\{0\}$.
+Back: Any finite family of hash functions.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 Tags: hashing::random hashing::universal
 <!--ID: 1721482558957-->

notes/lambda-calculus/index.md

@@ -432,7 +432,7 @@ END%%
 %%ANKI
 Basic
 How many occurences of $x$ are in $((xy)(\lambda x. (xy)))$?
-Back: Three.
+Back: $3$
 Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
 <!--ID: 1716743248125-->
 END%%

notes/proofs/index.md

@@ -1,3 +1,408 @@
 ---
 title: Proofs
+TARGET DECK: Obsidian::STEM
+FILE TAGS: proof::method
+tags:
+  - proof
 ---
+
+## Overview
+
+A **direct proof** is a sequence of statements, either givens or deductions of previous statements, whose last statement is the conclusion to be proved.
+
+%%ANKI
+Basic
+What is a direct proof?
+Back: A proof whose arguments follow directly one after another, up to the conclusion.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073057-->
+END%%
+
+%%ANKI
+Basic
+Generally speaking, what should the first statement of a direct proof be?
+Back: A hypothesis, if one exists.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073062-->
+END%%
+
+%%ANKI
+Basic
+Generally speaking, what should the last statement of a direct proof be?
+Back: The conclusion to be proved.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073065-->
+END%%
+
+An **indirect proof** works by assuming the denial of the desired conclusion leads to a contradiction in some way.
+
+%%ANKI
+Basic
+What is an indirect proof?
+Back: A proof in which the denial of a conclusion is assumed and shown to yield a contradiction.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073070-->
+END%%
+
+%%ANKI
+Cloze
+A {direct} proof is contrasted to an {indirect} proof.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073073-->
+END%%
+
+## Conditional Proofs
+
+A **conditional proof** is a proof method used to prove a conditional statement, i.e. statements of form: $$P_1 \land \cdots \land P_n \Rightarrow Q$$
+Note we can assume all the hypotheses are true since if one were false, the implication holds regardless. Direct proofs of the above form are called **conditional proofs** (CP).
+
+%%ANKI
+Basic
+What are conditional proofs?
+Back: Methods for proving propositions of form $P_1 \land \cdots \land P_n \Rightarrow Q$.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073076-->
+END%%
+
+%%ANKI
+Basic
+Which of conditional proofs or direct proofs is more general?
+Back: N/A.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073079-->
+END%%
+
+%%ANKI
+Basic
+Which of conditional proofs or indirect proofs is more general?
+Back: N/A.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073082-->
+END%%
+
+%%ANKI
+Basic
+Conditional proofs are used to solve propositions of what form?
+Back: $P_1 \land \cdots \land P_n \Rightarrow Q$
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073086-->
+END%%
+
+%%ANKI
+Basic
+*How* do we justify assuming the hypotheses in a conditional proof?
+Back: If any hypothesis were false, the conditional we are proving trivially holds.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073089-->
+END%%
+
+%%ANKI
+Basic
+Which proof method does CP stand for?
+Back: **C**onditional **p**roofs.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073092-->
+END%%
+
+### Proof by Contraposition
+
+Since a conditional and its contrapositive are logically equivalent, we can instead prove the negation of the conclusion leads to the negation of our hypotheses.
+
+%%ANKI
+Cloze
+{$P \Rightarrow Q$} is the contrapositive of {$\neg Q \Rightarrow \neg P$}.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073095-->
+END%%
+
+%%ANKI
+Basic
+Consider conditional $P \Rightarrow Q$. A proof by contrapositive typically starts with what assumption?
+Back: $\neg Q$
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073098-->
+END%%
+
+%%ANKI
+Basic
+How do you perform a proof by contraposition?
+Back: By showing the negation of the conclusion yields the negation of the hypotheses.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073101-->
+END%%
+
+%%ANKI
+Basic
+*Why* is proof by contraposition valid?
+Back: A conditional and its contrapositive are logically equivalent.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073104-->
+END%%
+
+%%ANKI
+Basic
+Is a proof by contraposition considered direct or indirect?
+Back: Indirect.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073108-->
+END%%
+
+### Proof by Contradiction
+
+To prove a proposition $P$ by contradiction, we assume $\neg P$ and derive a statement known to be false. Since mathematics is (in most cases) consistent, $P$ must be true.
+
+%%ANKI
+Basic
+Consider conditional $P \Rightarrow Q$. A proof by contradiction typically starts with what assumption?
+Back: $\neg P$
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073112-->
+END%%
+
+%%ANKI
+Basic
+What are the two most common indirect conditional proof strategies?
+Back: Proof by contraposition and proof by contradiction.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073116-->
+END%%
+
+%%ANKI
+Basic
+How do you perform a proof by contradiction?
+Back: Assume the negation of some statement and derive a contradiction.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073121-->
+END%%
+
+%%ANKI
+Basic
+*Why* is proof by contradiction valid?
+Back: It's assumed mathematics is consistent. If we prove a false statement, then our assumption is wrong.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073125-->
+END%%
+
+%%ANKI
+Basic
+Is a proof by contradiction considered direct or indirect?
+Back: Indirect.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073130-->
+END%%
+
+## Existence Proofs
+
+An **existence proof** is a proof method used to prove an existential statement, i.e. statements of form: $$\exists x, P(x)$$
+
+%%ANKI
+Basic
+What are existence proofs?
+Back: Methods for proving propositions of form $\exists x, P(x)$.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073134-->
+END%%
+
+%%ANKI
+Basic
+Which of existence proofs or direct proofs is more general?
+Back: N/A.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073137-->
+END%%
+
+%%ANKI
+Basic
+Which of existence proofs or indirect proofs is more general?
+Back: N/A.
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073140-->
+END%%
+
+%%ANKI
+Basic
+Existence proofs are used to solve propositions of what form?
+Back: $\exists x, P(x)$
+Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
+<!--ID: 1721824073143-->
+END%%
+
+An existence proof is said to be **constructive** if it demonstrates the existence of an object by creating (or providing a method for creating) the object. Otherwise it is said to be **non-constructive**.
+
+%%ANKI
+Basic
+Which more general proof method do constructive proofs fall under?
+Back: Existence proofs.
+Reference: “Constructive Proof,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Constructive_proof](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
+<!--ID: 1721824073146-->
+END%%
+
+%%ANKI
+Basic
+Is a constructive proof considered direct or indirect?
+Back: Usually direct.
+Reference: “Constructive Proof,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Constructive_proof](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
+<!--ID: 1721824073149-->
+END%%
+
+%%ANKI
+Basic
+Which more general proof method do non-constructive proofs fall under?
+Back: Existence proofs.
+Reference: “Constructive Proof,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Constructive_proof](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
+<!--ID: 1721824073152-->
+END%%
+
+%%ANKI
+Basic
+Is a non-constructive proof considered direct or indirect?
+Back: Usually indirect.
+Reference: “Constructive Proof,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Constructive_proof](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
+<!--ID: 1721824073155-->
+END%%
+
+## Induction
+
+Let $P(n)$ be a predicate. To prove $P(n)$ is true for all $n \geq n_0$, we prove:
+
+* **Base case**: Prove $P(n_0)$ is true. This is usually done directly.
+* **Inductive case**: Prove $P(k) \Rightarrow P(k + 1)$ for all $k \geq n_0$.
+
+Within the inductive case, $P(k)$ is known as the **inductive hypothesis**.
+
+%%ANKI
+Cloze
+The {base case} is to induction whereas {initial conditions} are to recursive definitions.
+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: 1714530152689-->
+END%%
+
+%%ANKI
+Cloze
+The {inductive case} is to induction whereas {recurrence relations} are to recursive definitions.
+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: 1714530152697-->
+END%%
+
+%%ANKI
+Basic
+What standard names are given to the cases in an induction proof?
+Back: The base case and inductive case.
+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: 1714530152701-->
+END%%
+
+%%ANKI
+Basic
+Let $(a_n)_{n \geq 0} = P(n)$ and $P(n) \Leftrightarrow n \geq 2$. How is $(a_n)$ written with terms expanded?
+Back: $F$, $F$, $T$, $T$, $T$, $\ldots$
+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: 1714530152705-->
+END%%
+
+%%ANKI
+Basic
+If proving $P(n)$ by weak induction, what are the first five terms of the underlying sequence?
+Back: $P(0)$, $P(1)$, $P(2)$, $P(3)$, $P(4)$, $\ldots$
+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: 1714530152709-->
+END%%
+
+%%ANKI
+Basic
+What proposition is typically proven in the base case of an inductive proof?
+Back: $P(n_0)$ for some $n_0 \geq 0$.
+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: 1714530152713-->
+END%%
+
+%%ANKI
+Basic
+What proposition is typically proven in the inductive case of an inductive proof?
+Back: $P(k) \Rightarrow P(k + 1)$ for all $k \geq n_0$.
+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: 1714530152718-->
+END%%
+
+%%ANKI
+Basic
+In weak induction, what special name is given to the antecedent of $P(k) \Rightarrow P(k + 1)$?
+Back: The inductive hypothesis.
+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: 1714530152722-->
+END%%
+
+%%ANKI
+Cloze
+{Closed formulas} are to recursive definitions as {direct proofs} are to proof strategies.
+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: 1714532476735-->
+END%%
+
+%%ANKI
+Cloze
+{Recurrence relations} are to recursive definitions as {induction} is to proof strategies.
+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: 1714532476742-->
+END%%
+
+%%ANKI
+Basic
+What proof strategy is most directly tied to recursion?
+Back: Induction.
+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: 1714574131911-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the inductive hypothesis of $P(n)$ using weak induction?
+Back: Assume $P(k)$ for some $k \geq n_0$.
+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: 1714574131942-->
+END%%
+
+### Strong Induction
+
+Strong induction expands the induction hypothesis. Let $P(n)$ be a predicate. To prove $P(n)$ is true for all $n \geq n_0$, we prove:
+
+* **Base case**: Prove $P(n_0)$ is true. This is usually done directly.
+* **Inductive case**: Assume $P(k)$ is true for all $n_0 \leq k < n$. Then prove $P(n)$ is true.
+
+%%ANKI
+Basic
+Using typical identifiers, what is the inductive hypothesis of $P(n)$ using strong induction?
+Back: Assume $P(k)$ for all $k < n$.
+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: 1714574131949-->
+END%%
+
+%%ANKI
+Basic
+Why is strong induction considered stronger than weak induction?
+Back: It can be used to solve at least the same set of problems weak induction can.
+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: 1714574131955-->
+END%%
+
+%%ANKI
+Basic
+What negation is introduced to explain why the strong induction assumption is valid?
+Back: If $P(n)$ is not true for all $n$, there exists a *first* $n_0$ for which $\neg P(n_0)$.
+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: 1714574131963-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes the base case of weak and strong induction proofs?
+Back: The latter may have more than one base case.
+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: 1714574131969-->
+END%%
+
+## Bibliography
+
+* “Constructive Proof,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Constructive_proof](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
+* 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).
+* Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

notes/proofs/induction.md

@@ -1,138 +0,0 @@
----
-title: Induction
-TARGET DECK: Obsidian::STEM
-FILE TAGS: algebra::sequence proof
-tags:
-  - proof
-  - sequence
----
-
-## Overview
-
-%%ANKI
-Cloze
-The {base case} is to induction whereas {initial conditions} are to recursive definitions.
-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: 1714530152689-->
-END%%
-
-%%ANKI
-Cloze
-The {inductive case} is to induction whereas {recurrence relations} are to recursive definitions.
-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: 1714530152697-->
-END%%
-
-%%ANKI
-Basic
-What standard names are given to the cases in an induction proof?
-Back: The base case and inductive case.
-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: 1714530152701-->
-END%%
-
-%%ANKI
-Basic
-Let $(a_n)_{n \geq 0} = P(n)$ and $P(n) \Leftrightarrow n \geq 2$. How is $(a_n)$ written with terms expanded?
-Back: $F$, $F$, $T$, $T$, $T$, $\ldots$
-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: 1714530152705-->
-END%%
-
-%%ANKI
-Basic
-If proving $P(n)$ by weak induction, what are the first five terms of the underlying sequence?
-Back: $P(0)$, $P(1)$, $P(2)$, $P(3)$, $P(4)$, $\ldots$
-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: 1714530152709-->
-END%%
-
-%%ANKI
-Basic
-What proposition is typically proven in the base case of an inductive proof?
-Back: $P(n_0)$ for some $n_0 \geq 0$.
-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: 1714530152713-->
-END%%
-
-%%ANKI
-Basic
-What proposition is typically proven in the inductive case of an inductive proof?
-Back: $P(k) \Rightarrow P(k + 1)$ for all $k \geq n_0$.
-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: 1714530152718-->
-END%%
-
-%%ANKI
-Basic
-In weak induction, what special name is given to the antecedent of $P(k) \Rightarrow P(k + 1)$?
-Back: The inductive hypothesis.
-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: 1714530152722-->
-END%%
-
-%%ANKI
-Cloze
-{Closed formulas} are to recursive definitions as {direct proofs} are to proof strategies.
-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: 1714532476735-->
-END%%
-
-%%ANKI
-Cloze
-{Recurrence relations} are to recursive definitions as {induction} is to proof strategies.
-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: 1714532476742-->
-END%%
-
-%%ANKI
-Basic
-What proof strategy is most directly tied to recursion?
-Back: Induction.
-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: 1714574131911-->
-END%%
-
-%%ANKI
-Basic
-Using typical identifiers, what is the inductive hypothesis of $P(n)$ using weak induction?
-Back: Assume $P(k)$ for some $k \geq n_0$.
-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: 1714574131942-->
-END%%
-
-%%ANKI
-Basic
-Using typical identifiers, what is the inductive hypothesis of $P(n)$ using strong induction?
-Back: Assume $P(k)$ for all $k < n$.
-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: 1714574131949-->
-END%%
-
-%%ANKI
-Basic
-Why is strong induction considered stronger than weak induction?
-Back: It can be used to solve at least the same set of problems weak induction can.
-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: 1714574131955-->
-END%%
-
-%%ANKI
-Basic
-What negation is introduced to explain why the strong induction assumption is valid?
-Back: If $P(n)$ is not true for all $n$, there exists a *first* $n_0$ for which $\neg P(n_0)$.
-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: 1714574131963-->
-END%%
-
-%%ANKI
-Basic
-What distinguishes the base case of weak and strong induction proofs?
-Back: The latter may have more than one base case.
-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: 1714574131969-->
-END%%
-
-## Bibliography
-
-* 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).

notes/set/relations.md

@@ -803,7 +803,49 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1720969371869-->
 END%%
 
-The set $[x]_R$ is defined by $[x]_R = \{t \mid xRt\}$. If $R$ is an equivalence relation and $x \in \mathop{\text{fld}}R$, then $[x]_R$ is called the **equivalence class of $x$ (modulo $R$)**. If the relation $R$ is fixed by the context, we may write just $[x]$.
+%%ANKI
+Basic
+The term "reflexive" is used to describe what kind of mathematical object?
+Back: Relations.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721693996250-->
+END%%
+
+%%ANKI
+Basic
+The term "symmetric" is used to describe what kind of mathematical object?
+Back: Relations.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721694448727-->
+END%%
+
+%%ANKI
+Cloze
+$R$ is symmetric iff {$R^{-1}$} {$\subseteq$} $R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721694448733-->
+END%%
+
+%%ANKI
+Basic
+The term "transitive" is used to describe what kind of mathematical object?
+Back: Relations.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721694448736-->
+END%%
+
+%%ANKI
+Cloze
+$R$ is transitive iff {$R \circ R$} {$\subseteq$} $R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721694448740-->
+END%%
+
+### Equivalence Classes
+
+The set $[x]_R$ is defined by $[x]_R = \{t \mid xRt\}$. If $R$ is an equivalence relation and $x \in \mathop{\text{fld}}R$, then $[x]_R$ is called the **equivalence class of $x$ (modulo $R$)**.
+
+If the relation $R$ is fixed by the context, we just write $[x]$.
 
 %%ANKI
 Basic
@@ -813,6 +855,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1721098094107-->
 END%%
 
+%%ANKI
+Basic
+How is set $[x]$ defined?
+Back: As $\{t \mid xRt\}$ for some unspecified $R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721697124837-->
+END%%
+
 %%ANKI
 Basic
 What is an equivalence class?
@@ -832,7 +882,7 @@ END%%
 %%ANKI
 Basic
 What kind of mathematical object is $R$ in $[x]_R$?
-Back: A relation.
+Back: A set.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721098094114-->
 END%%
@@ -847,7 +897,7 @@ END%%
 
 %%ANKI
 Cloze
-If {1:$R$ is an equivalence relation} and {1:$x \in \mathop{\text{fld} }R$}, then $[x]_R$ is called the {2:equivalence class of $x$} (modulo {2:$R$}).
+If {1:$R$ is an equivalence relation} and $x \in$ {2:$\mathop{\text{fld} }R$}, then $[x]_R$ is called the {2:equivalence class of $x$} (modulo {2:$R$}).
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721098094128-->
 END%%
@@ -891,7 +941,23 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1721098094158-->
 END%%
 
-## Partitions
+%%ANKI
+Basic
+Given sets $A$ and $x$, how can $[x]_A$ be rewritten as an image?
+Back: $A[\![\{x\}]\!]$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721696946316-->
+END%%
+
+%%ANKI
+Basic
+Given sets $A$ and $x$, how can we write $A[\![\{x\}]\!]$ more compactly?
+Back: $[x]_A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721696946369-->
+END%%
+
+### Partitions
 
 A **partition** $\Pi$ of a set $A$ is a set of nonempty subsets of $A$ that is disjoint and exhaustive.
 
@@ -1001,7 +1067,38 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1721136390215-->
 END%%
 
-## Quotient Sets
+%%ANKI
+Basic
+What name is given to a member of a partition of a set?
+Back: A cell.
+Reference: “Partition of a Set,” in _Wikipedia_, June 18, 2024, [https://en.wikipedia.org/w/index.php?title=Partition_of_a_set](https://en.wikipedia.org/w/index.php?title=Partition_of_a_set&oldid=1229656401).
+<!--ID: 1721696946377-->
+END%%
+
+%%ANKI
+Cloze
+Let $R$ be an equivalence relation. Then {1:cell} $[x]$ of partition $A / R$ is an {2:equivalence class of $A$} (modulo {2:$R$}).
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721696946384-->
+END%%
+
+%%ANKI
+Basic
+Let $R$ be the equivalence relation induced by partition $\Pi$ of $A$. What does $A / R$ equal?
+Back: $\Pi$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721728868200-->
+END%%
+
+%%ANKI
+Basic
+Let $R$ be an equivalence relation on $A$. What equivalence relation does partition $A / R$ induce?
+Back: $R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721728868210-->
+END%%
+
+### Quotient Sets
 
 If $R$ is an equivalence relation on $A$, then the **quotient set** "$A$ modulo $R$" is defined as $$A / R = \{[x]_R \mid x \in A\}.$$
 
@@ -1019,15 +1116,7 @@ END%%
 
 %%ANKI
 Basic
-Members of $A / R$ are called what?
+Quotient set $A / R$ is a partition of what set?
-Back: Equivalence classes.
-Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1721218408454-->
-END%%
-
-%%ANKI
-Basic
-$A / R$ is a partition of what set?
 Back: $A$
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721218408484-->
@@ -1041,6 +1130,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1721218408508-->
 END%%
 
+%%ANKI
+Basic
+Consider set $A / R$. What kind of mathematical object is $A$?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721698416717-->
+END%%
+
 %%ANKI
 Basic
 Consider quotient set $A / R$. What kind of mathematical object is $A$?
@@ -1049,6 +1146,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1721218408514-->
 END%%
 
+%%ANKI
+Basic
+Consider set $A / R$. What kind of mathematical object is $R$?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721698416723-->
+END%%
+
 %%ANKI
 Basic
 Consider quotient set $A / R$. What kind of mathematical object is $R$?
@@ -1057,10 +1162,18 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1721218408520-->
 END%%
 
+%%ANKI
+Basic
+How is set $A / R$ defined?
+Back: As $\{[x]_R \mid x \in A\}$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721698416727-->
+END%%
+
 %%ANKI
 Basic
 How is quotient set $A / R$ defined?
-Back: As set $\{[x]_R \mid x \in A\}$.
+Back: As $\{[x]_R \mid x \in A\}$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721218408525-->
 END%%
@@ -1131,4 +1244,5 @@ END%%
 ## Bibliography
 
 * “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
-* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+* “Partition of a Set,” in _Wikipedia_, June 18, 2024, [https://en.wikipedia.org/w/index.php?title=Partition_of_a_set](https://en.wikipedia.org/w/index.php?title=Partition_of_a_set&oldid=1229656401).

notes/set/trees.md

@@ -100,7 +100,7 @@ END%%
 Basic
 If the following isn't a forest, why not?
 ![[forest.png]]
-Back: N/A
+Back: N/A. It is.
 Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1711136844934-->
 END%%