From fc098cdaf2d40cf9defd7880a04180c559e6a434 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Thu, 2 May 2024 07:23:53 -0600 Subject: [PATCH] Induction and heaps. --- .../plugins/obsidian-to-anki-plugin/data.json | 20 +- notes/_journal/2024-04-30.md | 15 ++ notes/_journal/{ => 2024-04}/2024-04-29.md | 0 notes/_journal/2024-05-01.md | 13 + notes/_journal/2024-05-02.md | 11 + notes/algebra/sequences/delta-constant.md | 16 +- notes/algebra/sequences/index.md | 229 +++++++++++++++++- notes/algorithms/heaps.md | 6 +- notes/git/index.md | 125 ++++++++++ notes/programming/text-sub.md | 14 +- notes/proofs/index.md | 3 + notes/proofs/induction.md | 138 +++++++++++ 12 files changed, 559 insertions(+), 31 deletions(-) create mode 100644 notes/_journal/2024-04-30.md rename notes/_journal/{ => 2024-04}/2024-04-29.md (100%) create mode 100644 notes/_journal/2024-05-01.md create mode 100644 notes/_journal/2024-05-02.md create mode 100644 notes/proofs/index.md create mode 100644 notes/proofs/induction.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 4c7e3af..964e2c3 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -249,14 +249,14 @@ "filesystems/index.md": "cbd2b0290a3ba3b32abec4bd8bfefad5", "filesystems/cas.md": "d41c0d2e943adecbadd10a03fd1e4274", "git/objects.md": "c6b7e6a26666386790d25d4ece38175d", - "git/index.md": "83d2d95fc549d9e8436946c7bd058d15", + "git/index.md": "a4762d9f897f45ee13a2681828778820", "encoding/integer.md": "2ab21152b9468f547ebee496e03bc410", "_journal/2024-02-29.md": "f610f3caed659c1de3eed5f226cab508", "_journal/2024-02/2024-02-28.md": "7489377c014a2ff3c535d581961b5b82", "_journal/2024-03-01.md": "a532486279190b0c12954966cbf8c3fe", "_journal/2024-02/2024-02-29.md": "0e502a2c8baf90c2f12859b03f10b5a1", "algebra/sequences.md": "97c217823aacf8910a1a37bde694ecfe", - "algebra/sequences/index.md": "06df0fd12c2367995037ec4c26c51a53", + "algebra/sequences/index.md": "50de3bdd4d56af118cf3c8670ccf7fcc", "_journal/2024-03-02.md": "08c3cae1df0079293b47e1e9556f1ce1", "_journal/2024-03/2024-03-01.md": "70da812300f284df72718dd32fc39322", "algebra/sequences/triangular-numbers.md": "39a84ee317d3760a2eda7279c83e921a", @@ -359,7 +359,7 @@ "_journal/2024-04-16.md": "0bf6e2f2a3afab73d528cee88c4c1a92", "_journal/2024-04/2024-04-15.md": "256253b0633d878ca58060162beb7587", "algebra/polynomials.md": "6e20029b44fe0d0c4f35ef8ee4874d82", - "algebra/sequences/delta-constant.md": "8292ae72cd1f36c649f3e224f2c0d853", + "algebra/sequences/delta-constant.md": "70ceb29a3ec0ebe2b58af20107a5a2d3", "_journal/2024-04-19.md": "a293087860a7f378507a96df0b09dd2b", "_journal/2024-04/2024-04-18.md": "f6e5bee68dbef90a21ca92a846930a88", "_journal/2024-04/2024-04-17.md": "331423470ea83fc990c1ee1d5bd3b3f1", @@ -376,19 +376,25 @@ "_journal/2024-04/2024-04-23.md": "20514052da91b06b979cacb3da758837", "_journal/2024-04-25.md": "10c98531cb90a6bc940ea7ae3342f98b", "_journal/2024-04/2024-04-24.md": "4cb04e0dea56e0b471fc0e428471a390", - "algorithms/heaps.md": "9978cd8abebca41e8f123e0f8ea4c5c8", + "algorithms/heaps.md": "0cba4acb7667dcab80fa4e7778e86cc8", "_journal/2024-04-26.md": "3ce37236a9e09e74b547a4f7231df5f0", "_journal/2024-04/2024-04-25.md": "5a81123af29f8ebf0a0d28f820a3a52e", "_journal/2024-04-28.md": "46726bf76a594b987c63ba8b9b6d13d3", "_journal/2024-04/2024-04-27.md": "b0f3753821c232bf819b00fb49415bd0", "_journal/2024-04/2024-04-26.md": "3ce37236a9e09e74b547a4f7231df5f0", "algorithms/sorting/heapsort.md": "94ac936dac6c841b4d0c9b7df3eba0d3", - "_journal/2024-04-29.md": "f751826099e2c746605a4e562288d480", + "_journal/2024-04-29.md": "7888f4e9497c9d8bd6f4aa759d9abc4d", "_journal/2024-04/2024-04-28.md": "b34a9fe3bccb1f224b96ca00e78ad061", "programming/assertions.md": "cf7996590c9f5eefd6f7862c4f92132e", - "programming/text-sub.md": "003b8c32ae2f6767cb0d900f85196e67", + "programming/text-sub.md": "4bf5287353fc6571dd15be8f9f509bee", "programming/equiv-trans.md": "fbadd593688e364ea92f74f23e54bcfc", - "programming/index.md": "bb082325e269a95236aa6aff9307fe59" + "programming/index.md": "bb082325e269a95236aa6aff9307fe59", + "_journal/2024-04-30.md": "369f98b9d91de89cc1f4f581bc530c0d", + "_journal/2024-04/2024-04-29.md": "b4fa2fd62e1b4fe34c1f71dc1e9f5b0b", + "proofs/induction.md": "ab9e939efe32a4fb7ef2a0808eeb569f", + "proofs/index.md": "51a7bc4e30b7a6cc0d4c5712ad603448", + "_journal/2024-05-01.md": "959ff67fe3db585ba6a7b121d853bbac", + "_journal/2024-05-02.md": "962e63ca2f80179253aaf52e6b2aeaad" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-04-30.md b/notes/_journal/2024-04-30.md new file mode 100644 index 0000000..ce6a17d --- /dev/null +++ b/notes/_journal/2024-04-30.md @@ -0,0 +1,15 @@ +--- +title: "2024-04-30" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] Sheet Music (10 min.) +- [ ] Go (1 Life & Death Problem) +- [ ] Korean (Read 1 Story) +- [ ] Interview Prep (1 Practice Problem) +- [ ] Log Work Hours (Max 3 hours) + +* Notes and exercises for chapter 2.4 "Solving Recurrence Relations" of "Discrete Mathematics: An Open Introduction". + * Primarily focused on the [[algebra/sequences/index#Characteristic Roots|characteristic root technique]]. +* Read through chapter 2.5 "Induction" but want to create additional notes. \ No newline at end of file diff --git a/notes/_journal/2024-04-29.md b/notes/_journal/2024-04/2024-04-29.md similarity index 100% rename from notes/_journal/2024-04-29.md rename to notes/_journal/2024-04/2024-04-29.md diff --git a/notes/_journal/2024-05-01.md b/notes/_journal/2024-05-01.md new file mode 100644 index 0000000..4fb1a20 --- /dev/null +++ b/notes/_journal/2024-05-01.md @@ -0,0 +1,13 @@ +--- +title: "2024-05-01" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] Sheet Music (10 min.) +- [ ] Go (1 Life & Death Problem) +- [ ] Korean (Read 1 Story) +- [ ] Interview Prep (1 Practice Problem) +- [x] Log Work Hours (Max 3 hours) + +* Finished induction chapter of "Discrete Mathematics: An Open Introduction". Skimmed through chapter 3 "Symbolic Logic and Proofs". \ No newline at end of file diff --git a/notes/_journal/2024-05-02.md b/notes/_journal/2024-05-02.md new file mode 100644 index 0000000..603a7b6 --- /dev/null +++ b/notes/_journal/2024-05-02.md @@ -0,0 +1,11 @@ +--- +title: "2024-05-02" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] Sheet Music (10 min.) +- [ ] Go (1 Life & Death Problem) +- [ ] Korean (Read 1 Story) +- [ ] Interview Prep (1 Practice Problem) +- [ ] Log Work Hours (Max 3 hours) \ No newline at end of file diff --git a/notes/algebra/sequences/delta-constant.md b/notes/algebra/sequences/delta-constant.md index 3f590f6..edef8ef 100644 --- a/notes/algebra/sequences/delta-constant.md +++ b/notes/algebra/sequences/delta-constant.md @@ -44,14 +44,6 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% -%%ANKI -Basic -What is the base case of the recursive definition of the $k$th differences of $(a_n)$? -Back: $k = 0$ -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 is the recurrence of the recursive definition of the $(k + 1)$st differences of $(a_n)$? @@ -92,6 +84,14 @@ Tags: calculus END%% +%%ANKI +Cloze +{Derivatives} are to continuous whereas {differences} are to discrete. +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). +Tags: calculus + +END%% + %%ANKI Basic What kind of mathematical expression do $\Delta^k$-constant sequences relate to? diff --git a/notes/algebra/sequences/index.md b/notes/algebra/sequences/index.md index 68a9de6..3d353bd 100644 --- a/notes/algebra/sequences/index.md +++ b/notes/algebra/sequences/index.md @@ -169,13 +169,9 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% -## Solving Recurrence Relations +## Recurrence Relations -We use three different strategies for solving recurrences: - -* Telescoping -* Iteration -* Characteristic Polynomials +To solve a recurrence relation means to find a closed form for the relation (with respect to initial conditions). %%ANKI Basic @@ -193,6 +189,10 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% +### Telescoping + +Telescoping refers to the property of summations in which consecutive terms cancel out. We can use telescoping to solve recurrences of form $a_n = a_{n-1} + f(n)$ by noticing that: $$\begin{align*} a_1 - a_0 & = f(1) \\ a_2 - a_1 & = f(2) \\ \vdots \\ a_n - a_{n-1} & = f(n) \\ \hline a_n - a_0 & = \sum_{k=1}^n f(n) \end{align*}$$ + %%ANKI Basic What does it mean for a sum to be telescoping? @@ -278,6 +278,10 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% +### Iteration + +Iteration refers to the expansion of terms, starting at the initial conditions, in the hope of discovering a pattern. It is more general than [[#Telescoping]] is. Consider $a_n = a_{n-1} + f(n)$ again. We solve with iteration like so: $$\begin{align*} a_1 & = a_0 + f(1) \\ a_2 & = (a_0 + f(1)) + f(2) \\ \vdots \\ a_n & = (\cdots(a_0 + f(1)) + f(2)) + \cdots) + f(n) \\ \hline a_n & = a_0 + \sum_{k=1}^n f(n) \end{align*}$$ + %%ANKI Basic What does it mean to solve a recurrence relation using iteration? @@ -343,6 +347,219 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% +### Characteristic Roots + +When encountering **linear homogeneous recurrence relations with constant coefficients**, we can use the characteristic root technique to solve. We demonstrate with a quadratic **characteristic polynomial**, though this technique generalizes to higher-order polynomials as well. + +Given recurrence relation $a_n + \alpha a_{n-1} + \beta a_{n-2} = 0$, the characteristic polynomial is $r^2 + \alpha r + \beta$. If $r_1$ and $r_2$ are distinct roots of the characteristic polynomial, then the solution to the recurrence relation is $$a_n = ar_1^n + br_2^n$$ +where $a$ and $b$ are determined by the initial conditions. If the characteristic polynomial only has one root $r$, the solution is instead $$a_n = ar^n + bnr^n$$ + +%%ANKI +Basic +The characteristic root technique only works when solving what kind of recurrence relation? +Back: Linear homogeneous recurrence relations with constant coefficients. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What does "linear" refer to in "linear homogeneous recurrence relations with constant coefficients"? +Back: The recurrence relation is a *linear* combination of previous terms. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What does "homogeneous" refer to in "linear homogeneous recurrence relations with constant coefficients"? +Back: *Every* term in the relation is a multiple of previous terms. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Why isn't $a_n = 2a_{n-1} + 3a_{n-2}$ a linear homogeneous recurrence relation with constant coefficients? +Back: It is. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Why isn't $a_n = 2a_{n-1} \cdot 3a_{n-2}$ a linear homogeneous recurrence relation with constant coefficients? +Back: It is non-linear. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Why isn't $a_n = 2a_{n-1} + 3$ a linear homogeneous recurrence relation with constant coefficients? +Back: It is non-homogeneous. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Why isn't $a_n = 2a_{n-1} + na_{n-2}$ a linear homogeneous recurrence relation with constant coefficients? +Back: It has a nonconstant coefficient. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What is the most famous linear homoegeneous recurrence relation with constant coefficients? +Back: The Fibonacci sequence's recurrence relation. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Why might we guess the solution to e.g. $a_n = a_{n-1} + 6a_{n-2}$ is geometric? +Back: Every step of iteration multiplies a previous iteration by $6$. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +How is $a_n = a_{n-1} + 6a_{n-2}$ factored to yield its characteristic polynomial? +Back: $r^{n-2}(r^2 - r - 6) = 0$ +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What is the characteristic *equation* of $a_n = a_{n-1} + 6a_{n-2}$? +Back: $r^2 - r - 6 = 0$ +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What is the characteristic *polynomial* of $a_n = a_{n-1} + 6a_{n-2}$? +Back: $r^2 - r - 6$ +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What is the characteristic *polynomial* of $a_n + \alpha a_{n-1} + \beta a_{n-2} = 0$? +Back: $r^2 + \alpha r + \beta$ +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What is the characteristic *equation* of $a_n + \alpha a_{n-1} + \beta a_{n-2} = 0$? +Back: $r^2 + \alpha r + \beta = 0$ +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What is the characteristic *equation* of $a_n = \alpha a_{n-1} + \beta a_{n-2}$? +Back: $r^2 - \alpha r - \beta = 0$ +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Given recurrence $a_n = \alpha a_{n-1} + \beta a_{n-2}$, what guess is used to derive the concept of a characteristic polynomial? +Back: The guessing of a geometric solution, e.g. $r^n$. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +What distinguishes the characteristic polynomial from the characteristic equation of a recurrence relation? +Back: The latter sets the characteristic polynomial equal to $0$. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Using the characteristic root technique, what determines the form of the closed solution? +Back: The number of distinct roots of the characteristic polynomial. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Using the characteristic root technique, what determines the form of the closed solution? +Back: The number of distinct roots of the characteristic polynomial. +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). +Tags: algebra::polynomial +END%% + +%%ANKI +Basic +Suppose the characteristic polynomial of $a_n = \alpha a_{n-1} + \beta a_{n-2}$ has distinct roots $r_1$ and $r_2$. What is its solution? +Back: $a_n = ar_1^n + br_2^n$ where $a$ and $b$ are determined by initial conditions. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +Suppose the characteristic polynomial of $a_n = \alpha a_{n-1} + \beta a_{n-2}$ has single root $r$. What is its solution? +Back: $a_n = ar^n + bnr^n$ where $a$ and $b$ are determined by initial conditions. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +If $a_n = \alpha a_{n-1} + \beta a_{n-2}$ has solution $a_n = ar^n + bnr^n$, how many roots does its characteristic polynomial have? +Back: One. +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). +Tags: algebra::polynomial + +END%% + +%%ANKI +Basic +If $a_n = \alpha a_{n-1} + \beta a_{n-2}$ has solution $a_n = ar_1^n + br_2^n$, how many roots does its characteristic polynomial have? +Back: Two. +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). +Tags: algebra::polynomial + +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). diff --git a/notes/algorithms/heaps.md b/notes/algorithms/heaps.md index 65b3a62..1fa1e36 100644 --- a/notes/algorithms/heaps.md +++ b/notes/algorithms/heaps.md @@ -249,7 +249,7 @@ END%% %%ANKI Basic -When is `Min_HEAPIFY_DOWN` a no-op? +When is `MIN_HEAPIFY_DOWN` a no-op? Back: When the current node is already smaller than both its children. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). @@ -297,7 +297,7 @@ END%% %%ANKI Basic -What is the height of a binary heap? +What is the height of a binary heap defined? Back: The height of the heap's root when viewed as a complete binary tree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). @@ -425,7 +425,7 @@ END%% %%ANKI Basic Why does `BUILD_MIN_HEAP` "ignore" the external nodes of a heap? -Back: Because they are already max-heaps of size $1$. +Back: Because they are already min-heaps of size $1$. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% diff --git a/notes/git/index.md b/notes/git/index.md index 02211f4..c9d775f 100644 --- a/notes/git/index.md +++ b/notes/git/index.md @@ -1,3 +1,128 @@ --- title: Git +TARGET DECK: Obsidian::STEM +FILE TAGS: git +tags: + - git --- + +## Overview + +Files in the working directory are in one of two states: + +* **Tracked** - files that were in the last snapshot or newly staged. +* **Untracked** - files in the working directory that aren't tracked. + +Tracked files may be unmodified, modified, or staged. + +%%ANKI +Basic +Files in the working directory are in one of what two states? +Back: Tracked and untracked. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +What is a tracked file? +Back: A file that is staged or exists in the latest snapshot. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +What is an untracked file? +Back: A file that is neither staged nor exists in the latest snapshot. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +What three statuses can a tracked file be in? +Back: Unmodified, modified, and staged. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +What does it mean for a tracked file to be unmodified? +Back: The version in the working directory is the same as in the last snapshot. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +What does it mean for a tracked file to be modified? +Back: The version in the working directory is different from that in the last snapshot. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +What does it mean for a tracked file to be staged? +Back: The version in the working directory has been added to the index. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +How do you convert an untracked file to a tracked file? +Back: Stage the file. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +What operation converts an unmodified file to a modified file? +Back: Editing the file. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Basic +You can convert an untracked file to a tracked file with what status? +Back: Staged. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Cloze +An {1:unmodified} tracked file becomes a {2:modified} tracked file after {2:editing} the file. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Cloze +A {1:modified} tracked file becomes a {2:staged} tracked file after {2:staging} the file. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Cloze +A {1:staged} tracked file becomes an {2:unmodified} tracked file after {2:committing}. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +%%ANKI +Cloze +An untracked file becomes a {1:staged} tracked file after {1:adding} the file. +Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). + +END%% + +## Bibliography + +* Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014). \ No newline at end of file diff --git a/notes/programming/text-sub.md b/notes/programming/text-sub.md index bdf507c..7f6539a 100644 --- a/notes/programming/text-sub.md +++ b/notes/programming/text-sub.md @@ -436,7 +436,7 @@ END%% %%ANKI Basic -Given valid expression $(b; [i]{\circ}s{:}e))$, what can be said about $i$? +Given valid expression $(b; [i]{\circ}s{:}e)$, what can be said about $i$? Back: $i$ is in the domain of $b$. Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. @@ -444,7 +444,7 @@ END%% %%ANKI Basic -Given valid expression $(b; [i]{\circ}s{:}e))$, what is the type of $b$? +Given valid expression $(b; [i]{\circ}s{:}e)$, what is the type of $b$? Back: A function. Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. @@ -452,7 +452,7 @@ END%% %%ANKI Basic -Given valid expression $(b; \epsilon{\circ}s{:}e))$, what is the type of $b$? +Given valid expression $(b; \epsilon{\circ}s{:}e)$, what is the type of $b$? Back: A scalar or function. Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. @@ -461,7 +461,7 @@ END%% %%ANKI Basic What is the base case of selector update syntax? -Back: Updates involving the null selector. +Back: $(b; \epsilon{:}g) = g$ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. END%% @@ -476,7 +476,7 @@ END%% %%ANKI Basic -The nuil selector is the identity element of what operation? +The null selector is the identity element of what operation? Back: Concatenation of sequences of subscripts. Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. @@ -525,7 +525,7 @@ END%% %%ANKI Basic Let $b$ be an array. How is $b[i][j] := e$ rewritten using selector update syntax? -Back: $(b; [i][j]{:}e)$ +Back: $b := (b; [i][j]{:}e)$ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. END%% @@ -556,7 +556,7 @@ END%% %%ANKI Basic -Maintaining selector update syntax, how is $(c; 1{:}3)[1]$ rewritten with a selector? +Maintaining selector update syntax, how is $(c; 1{:}3)[1]$ more explicitly written with a selector? Back: $(c; [1]{:}3)[1]$ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. diff --git a/notes/proofs/index.md b/notes/proofs/index.md new file mode 100644 index 0000000..87752fd --- /dev/null +++ b/notes/proofs/index.md @@ -0,0 +1,3 @@ +--- +title: Proofs +--- diff --git a/notes/proofs/induction.md b/notes/proofs/induction.md new file mode 100644 index 0000000..4e340d0 --- /dev/null +++ b/notes/proofs/induction.md @@ -0,0 +1,138 @@ +--- +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: $F$, $F$, $F$, $T$, $T$, $\ldots$ +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 forms} are to recursive definitions as {direct} is to proofs. +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 proofs. +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 contradiction 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