Notes on proof methods.
parent
5dc598393e
commit
0eeb72b9ec
|
|
@ -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": [
|
||||
|
|
|
|||
|
|
@ -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]].
|
||||
|
|
@ -10,3 +10,4 @@ title: "2024-07-22"
|
|||
|
||||
* Beginning notes on [[natural-deduction|natural deduction]].
|
||||
* Read chapter 3 "Overview of the Term Sheet" of "Venture Deals".
|
||||
* Finished "Equivalence Relations" exercises of "Elements of Set Theory".
|
||||
|
|
@ -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).
|
||||
|
|
@ -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
|
||||
|
|
|
|||
|
|
@ -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).
|
||||
<!--ID: 1718069881709-->
|
||||
END%%
|
||||
|
|
|
|||
|
|
@ -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?
|
||||
Back: $\forall$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1707494819971-->
|
||||
What non-counting quantifer denotes unique existential quantification?
|
||||
Back: $\exists!$
|
||||
Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
|
||||
<!--ID: 1721824073159-->
|
||||
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-->
|
||||
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.
|
||||
<!--ID: 1721824073162-->
|
||||
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-->
|
||||
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.
|
||||
<!--ID: 1721824073165-->
|
||||
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-->
|
||||
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.
|
||||
<!--ID: 1721824073168-->
|
||||
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-->
|
||||
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.
|
||||
<!--ID: 1721824073172-->
|
||||
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
|
|||
<!--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**).
|
||||
|
|
@ -315,3 +360,4 @@ END%%
|
|||
|
||||
* 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).
|
||||
* Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
|
||||
|
|
@ -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
|
|||
<!--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.
|
||||
|
|
|
|||
|
|
@ -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-->
|
||||
|
|
|
|||
|
|
@ -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%%
|
||||
|
|
|
|||
|
|
@ -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.
|
||||
|
|
@ -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).
|
||||
|
|
@ -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?
|
||||
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?
|
||||
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).
|
||||
<!--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%%
|
||||
|
|
@ -1132,3 +1245,4 @@ END%%
|
|||
|
||||
* “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).
|
||||
* “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).
|
||||
|
|
@ -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%%
|
||||
|
|
|
|||
Loading…
Reference in New Issue