diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index ce010c4..bea4fdc 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -251,7 +251,7 @@ "combinatorics/inclusion-exclusion.md": "c27b49ee03cc5ee854d0e8bd12a1d505", "_journal/2024-02-21.md": "b9d944ecebe625da5dd72aeea6a916a2", "_journal/2024-02/2024-02-20.md": "af2ef10727726200c4defe2eafc7d841", - "algebra/radices.md": "81c887836b38a8584234a74d612aef12", + "algebra/radices.md": "662a3e2f611dd9655f1ef6c7ed0a9943", "_journal/2024-02-22.md": "e01f1d4bd2f7ac2a667cdfd500885a2a", "_journal/2024-02/2024-02-21.md": "f423137ae550eb958378750d1f5e98c7", "_journal/2024-02-23.md": "219ce9ad15a8733edd476c97628b71fd", @@ -345,7 +345,7 @@ "x86-64/declarations.md": "75bc7857cf2207a40cd7f0ee056af2f2", "x86-64/instructions.md": "06b7fbe1a7a9568b80239310eb72e87a", "git/refs.md": "e20c2c9b14ba6c2bd235416017c5c474", - "set/trees.md": "ea3818b3750838b2d273e01874cf300f", + "set/trees.md": "f21182d92a480613f18c0a9d8839fb57", "_journal/2024-03-24.md": "1974cdb9fc42c3a8bebb8ac76d4b1fd6", "_journal/2024-03/2024-03-23.md": "ad4e92cc2bf37f174a0758a0753bf69b", "_journal/2024-03/2024-03-22.md": "a509066c9cd2df692549e89f241d7bd9", @@ -416,8 +416,8 @@ "programming/index.md": "bb082325e269a95236aa6aff9307fe59", "_journal/2024-04-30.md": "369f98b9d91de89cc1f4f581bc530c0d", "_journal/2024-04/2024-04-29.md": "b4fa2fd62e1b4fe34c1f71dc1e9f5b0b", - "proofs/induction.md": "25b195c80df87aac399cf1234389ef9e", - "proofs/index.md": "51a7bc4e30b7a6cc0d4c5712ad603448", + "proofs/induction.md": "36ab5a92ae3cf9bb2def333dc41d79ff", + "proofs/index.md": "ab48e1812838365767eaca2e95611b48", "_journal/2024-05-01.md": "959ff67fe3db585ba6a7b121d853bbac", "_journal/2024-05-02.md": "d7d6ba7e065d807986f0bd77281c0bb1", "data-structures/priority-queues.md": "8c5c6bf62b1a39d8f1f72b800fcb17ff", @@ -462,7 +462,7 @@ "_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c", "_journal/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", "hashing/direct-addressing.md": "f75cc22e74ae974fe4f568a2ee9f951f", - "hashing/index.md": "ee4335b307ff1dc740789e9972b19e50", + "hashing/index.md": "b260890abe8ef5cde8bd0804c58c341e", "set/classes.md": "6776b4dc415021e0ef60b323b5c2d436", "_journal/2024-05-19.md": "fddd90fae08fab9bd83b0ef5d362c93a", "_journal/2024-05/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", @@ -485,7 +485,7 @@ "_journal/2024-05/2024-05-25.md": "3e8a0061fa58a6e5c48d12800d1ab869", "_journal/2024-05-27.md": "b36636d10eab34380f17f288868df3ae", "_journal/2024-05/2024-05-26.md": "abe84b5beae74baa25501c818e64fc95", - "algebra/set.md": "fe0121964ae8c788a2afb6031b4086d9", + "algebra/set.md": "f7a775cd9f6bf0b3a127fc9c3b9a3c15", "algebra/boolean.md": "ee41e624f4d3d3aca00020d9a9ae42c8", "git/merge-conflicts.md": "761ad6137ec51d3877f7d5b3615ca5cb", "_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337", @@ -510,13 +510,13 @@ "_journal/2024-06/2024-06-04.md": "52b28035b9c91c9b14cef1154c1a0fa1", "_journal/2024-06-06.md": "3f9109925dea304e7172df39922cc95a", "_journal/2024-06/2024-06-05.md": "b06a0fa567bd81e3b593f7e1838f9de1", - "set/relations.md": "29f9b8220cc147ae638e4832c0e82919", + "set/relations.md": "6033e9506744dc845da6b36856aaf45c", "_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2", "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381", "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8", "_journal/2024-06/2024-06-07.md": "c6bfc4c1e5913d23ea7828a23340e7d3", "lambda-calculus/alpha-conversion.md": "007828faf9b4ace5bd30b87a36a90dcf", - "lambda-calculus/index.md": "64efe9e4f6036d3f5b4ec0dc8cd3e7b9", + "lambda-calculus/index.md": "76d58f85c135c7df00081f47df31168e", "x86-64/instructions/condition-codes.md": "1f59f0b81b2e15582b855d96d1d377da", "x86-64/instructions/logical.md": "818428b9ef84753920dc61e5c2de9199", "x86-64/instructions/arithmetic.md": "271218d855e7291f119f96e91f582738", @@ -627,16 +627,19 @@ "formal-system/proof-system/index.md": "1c95481cbb2e79ae27f6be1869599657", "formal-system/proof-system/equiv-trans.md": "4d5e9236944c3ea99f484bfcb14292d0", "formal-system/logical-system/index.md": "708bb1547e7343c236068c18da3f5dc0", - "formal-system/logical-system/pred-logic.md": "2524ccc09561bc219dab3f32010a0161", + "formal-system/logical-system/pred-logic.md": "6db7f2a3734b6f3d48313410dc611bd5", "formal-system/logical-system/prop-logic.md": "b61ce051795d5a951c763b928ec5cea8", "formal-system/index.md": "3d31c99bffdcb05de9f2e32ac6319952", "programming/short-circuit.md": "c256ced42dc3b493aff5a356e5383b6e", "formal-system/abstract-rewriting.md": "8424314a627851c5b94be6163f64ba30", - "_journal/2024-07-22.md": "dbbf1666c0ed939ce0ce339d41231b04", + "_journal/2024-07-22.md": "d2ca7ce0bbeef76395fee33c9bf36e9d", "_journal/2024-07/2024-07-21.md": "62c2651999371dd9ab10d964dac3d0f8", - "formal-system/proof-system/natural-deduction.md": "307f4c24526311a209d0686065ae601c", - "startups/term-sheet.md": "3b7fe2e4b067da47cdb2f0517c10e73f", - "startups/financing-rounds.md": "00a622fda2b4b442901bde2842309088" + "formal-system/proof-system/natural-deduction.md": "62db68d3cfbda84426f390ddd4a16a54", + "startups/term-sheet.md": "6b6152af78addb3fe818a7fc9d375fbf", + "startups/financing-rounds.md": "00a622fda2b4b442901bde2842309088", + "_journal/2024-07-23.md": "35e18a1d9a8dd0a97e1d9898bc1d8f01", + "_journal/2024-07/2024-07-22.md": "8170a92496c2c5374fc3411bddf3b17d", + "_journal/2024-07-24.md": "e7f4b617435e528b00a241b26fde1ce9" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-07-24.md b/notes/_journal/2024-07-24.md new file mode 100644 index 0000000..e870c9f --- /dev/null +++ b/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]]. \ No newline at end of file diff --git a/notes/_journal/2024-07-22.md b/notes/_journal/2024-07/2024-07-22.md similarity index 77% rename from notes/_journal/2024-07-22.md rename to notes/_journal/2024-07/2024-07-22.md index cab6e5b..cde920b 100644 --- a/notes/_journal/2024-07-22.md +++ b/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". \ No newline at end of file +* Read chapter 3 "Overview of the Term Sheet" of "Venture Deals". +* Finished "Equivalence Relations" exercises of "Elements of Set Theory". \ No newline at end of file diff --git a/notes/_journal/2024-07/2024-07-23.md b/notes/_journal/2024-07/2024-07-23.md new file mode 100644 index 0000000..370b2fa --- /dev/null +++ b/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). \ No newline at end of file diff --git a/notes/algebra/radices.md b/notes/algebra/radices.md index 2888031..87acee9 100644 --- a/notes/algebra/radices.md +++ b/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 diff --git a/notes/algebra/set.md b/notes/algebra/set.md index bd18db5..afd6de9 100644 --- a/notes/algebra/set.md +++ b/notes/algebra/set.md @@ -293,8 +293,8 @@ END%% %%ANKI Basic -Under what two conditions is $A \times B = B \times A$? -Back: $A = B$ or either set is the empty set. +Under what three conditions is $A \times B = B \times A$? +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). END%% diff --git a/notes/formal-system/logical-system/pred-logic.md b/notes/formal-system/logical-system/pred-logic.md index 76d248e..d90fb6f 100644 --- a/notes/formal-system/logical-system/pred-logic.md +++ b/notes/formal-system/logical-system/pred-logic.md @@ -114,64 +114,52 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in 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? -Back: $\forall$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - +What non-counting quantifer denotes unique existential quantification? +Back: $\exists!$ +Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d. + 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. - +Unique existential quantification can be expressed using what counting quantification? +Back: $\exists^{=1}$ +Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d. + 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. - +How is $\exists! x, P(x)$ expanded using the basic existential and universal quantifiers? +Back: $(\exists x, P(x)) \land (\forall x, \forall y, (P(x) \land P(y)) \Rightarrow (x = y))$ +Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d. + 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. - -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. - +How do we write the equivalent existence (not uniqueness) assertion made by $\exists! x, P(x)$? +Back: $\exists x, P(x))$ +Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d. + 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. - +How do we write the equivalent uniqueness (not existence) assertion made by $\exists! x, P(x)$? +Back: $\forall x, \forall y, (P(x) \land P(y)) \Rightarrow (x = y)$ +Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d. + 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%% - -### Counting +#### 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 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. + +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. + +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. + +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. + +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. + +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. + +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). \ No newline at end of file +* 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. \ No newline at end of file diff --git a/notes/formal-system/proof-system/natural-deduction.md b/notes/formal-system/proof-system/natural-deduction.md index d8b7974..839c0ee 100644 --- a/notes/formal-system/proof-system/natural-deduction.md +++ b/notes/formal-system/proof-system/natural-deduction.md @@ -181,8 +181,8 @@ END%% ### Implication -For propositions $E1$ and $E2$, $${\Rightarrow}{\text{-}}I: \quad \text{TODO}$$ -and $${\Rightarrow}{\text{-}}E: \quad \begin{array}{c} E1 \Rightarrow E2, E1 \\ \hline E2 \end{array}$$ +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 $${\Rightarrow}{\text{-}}E: \quad \begin{array}{c} E_1 \Rightarrow E_2, E_1 \\ \hline E_2 \end{array}$$ %%ANKI Basic @@ -192,6 +192,14 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in 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. + +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 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. diff --git a/notes/hashing/index.md b/notes/hashing/index.md index 023f987..a88acd4 100644 --- a/notes/hashing/index.md +++ b/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 diff --git a/notes/lambda-calculus/index.md b/notes/lambda-calculus/index.md index 336ce54..5b984d1 100644 --- a/notes/lambda-calculus/index.md +++ b/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). END%% diff --git a/notes/proofs/index.md b/notes/proofs/index.md index 87752fd..dec1428 100644 --- a/notes/proofs/index.md +++ b/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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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. \ No newline at end of file diff --git a/notes/proofs/induction.md b/notes/proofs/induction.md deleted file mode 100644 index 34eaece..0000000 --- a/notes/proofs/induction.md +++ /dev/null @@ -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). - -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). \ No newline at end of file diff --git a/notes/set/relations.md b/notes/set/relations.md index a0f8e23..5c2ba29 100644 --- a/notes/set/relations.md +++ b/notes/set/relations.md @@ -803,7 +803,49 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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). + +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). + +END%% + +%%ANKI +Cloze +$R$ is symmetric iff {$R^{-1}$} {$\subseteq$} $R$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +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). + +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). + +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 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). + +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). 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). END%% @@ -891,7 +941,23 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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). + +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). + +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 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). + +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). + +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). + +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). + +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? -Back: Equivalence classes. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - -%%ANKI -Basic -$A / R$ is a partition of what set? +Quotient set $A / R$ is a partition of what set? Back: $A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). @@ -1041,6 +1130,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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). + +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 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). + +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 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). + +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). 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). \ No newline at end of file +* 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). \ No newline at end of file diff --git a/notes/set/trees.md b/notes/set/trees.md index 3b6e198..9d6f885 100644 --- a/notes/set/trees.md +++ b/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). END%%