From de2f686c1688c03f61bae66424ca153213241642 Mon Sep 17 00:00:00 2001 From: JJ Date: Thu, 18 Jan 2024 13:46:20 -0800 Subject: meow --- _layouts/default.html | 10 +++++++ mathematics/algebra.md | 75 ++++++++++++++++++++++++++++++++++++++++++++++++++ mathematics/logic.md | 14 +++++----- web/html.md | 26 ++++++++--------- 4 files changed, 105 insertions(+), 20 deletions(-) diff --git a/_layouts/default.html b/_layouts/default.html index 4583528..ad4c7ed 100644 --- a/_layouts/default.html +++ b/_layouts/default.html @@ -7,6 +7,16 @@ + + + diff --git a/mathematics/algebra.md b/mathematics/algebra.md index 5838147..5434ff9 100644 --- a/mathematics/algebra.md +++ b/mathematics/algebra.md @@ -4,3 +4,78 @@ title: mathematics/algebra --- # algebra + +modern algebra is the study of **algebraic structures**: groups, rings, fields, modules, vector spaces, lattices, and the like. + +these structures are very general: and so results from abstract algebra can be applied to a wide variety of situations. + +## structures + +An **algebraic structure** is a set with a collection of *operations* and a finite set of *axioms* those operations must satisfy. + +A **group** $G$ is a set with a single binary operation ⋆ satisfying the following axioms: +- associativity: $∀a,b,c : (a⋆b)⋆c = a⋆(b⋆c) +- identity: $∃e, ∀a : e⋆a = a⋆e = a$ +- inverse: $∀a, ∃a^{-1} : a⋆a^{-1} = e$ +- An Abelian or **commutative group** satisfies an additional axiom: + - commutativity: $∀a,b : a⋆b=b⋆a$ + +A **monoid** is a group without an inverse operation. + +A **ring** $R$ is a set with two binary operations + and × satisfying the following axioms: +- $(R, +)$ is a *commutative group*: + - associativity: $∀a,b,c : (a+b)+c = a+(b+c) + - additive identity: $∃0, ∀a : 0+a = a+0 = a$ + - additive inverse: $∀a, ∃-a : a+(-a) = 0$ + - commutativity: $∀a,b : a+b=b+a$ +- $(R, ×)$ is a *monoid* + - associativity: $∀a,b,c : (a×b)×c = a×(b×c) + - multiplicative identity: $∃1, ∀a : 1×a = a×1 = a$ +- The *distributive laws* hold for + and ×: + - $∀a,b,c : (a+b) × c = (a×c)+(b×c)$ + - $∀a,b,c : a × (b+c) = (a×b) + (a×c)$ +- An Abelian or **commutative ring** satisfies an additional axiom: + - commutativity (of ×): $∀a,b : a×b=b×a$ + +A **field** is a *commutative ring* where $0 ≠ 1$ and all elements sans $0$ have an inverse $a^{-1}$ under multiplication. + +A **vector space** $V$ over a field $F$ of scalars is a set with a binary operation + and a binary function satisfying the following axioms: +- $(V, +)$ is a *commutative group*: + - associativity: $∀u,v,w : (u+v)+w = u+(v+w) + - additive identity: $∃0, ∀v: 0+v = v+0 = v$ + - additive inverse: $∀v, ∃-v: v+(-v) = 0$ + - commutativity: $∀u,v : u+v=v+u$ +- $(V, )$ is a *scalar operation*: + - scalar identity: $∃1 ∈ F : 1v = v1 = v$ + - commutativity: $∀a,b ∈ F, ∀v ∈ V (ab)v = a(bv)$ +- The *distributive laws* hold: + - $∀a ∈ F, ∀u,v ∈ V : a(u+v) = au+av$ + - $∀a,b ∈ F, ∀v ∈ V : (a+b)v = av + bv$ + +A **module** $M$ is a generalization of a *vector space* to function over a ring $R$ instead of a field. + +A **lattice** $L$ is a set with two binary operations ∧ and ∨ satisfying the following axioms: +- commutativity: + - $∀a,b : a ∧ b = b ∧ a$ + - $∀a,b : a ∨ b = b ∨ a$ +- associativity: + - $∀a,b,c : a ∧ (b ∧ c) = (a ∧ b) ∧ c$ + - $∀a,b,c : a ∨ (b ∨ c) = (a ∨ b) ∨ c$ +- absorption: + - $∀a,b : a ∧ (a ∨ b) = a$ + - $∀a,b : a ∨ (a ∧ b) = a$ +- idempotence: + - $∀a : a ∧ a = a$ + - $∀a : a ∨ a = a$ + +## group theory + +## ring theory + +## galois theory + +## linear algebra + +## order theory + +a lattice may alternatively be defined as... diff --git a/mathematics/logic.md b/mathematics/logic.md index e19cb52..0760a8c 100644 --- a/mathematics/logic.md +++ b/mathematics/logic.md @@ -18,9 +18,9 @@ and many others. ## orders of logic -### [propositional logic](https://ncatlab.org/nlab/show/propositional+logic) +### propositional logic -**propositional logic** or **zeroth-order logic** deals with raw *propositions*. +[**propositional logic**](https://ncatlab.org/nlab/show/propositional+logic) or **zeroth-order logic** deals with raw *propositions*. **propositions** are statements that *reduce* to a **truth value**. truth values are classically either true or false. in non-classical logics, this can differ. @@ -28,11 +28,11 @@ the basic foundations of propositional logic are as follows: notation | definition ---------|-------------- +0 | *false* +1 | *true* p | a *proposition* ¬p | *not* p p → q | *if* p *then* q, p *implies* q -0 | *false* -1 | *true* several logical connectives are *derivable* from the above: @@ -40,17 +40,17 @@ notation | derivation | definition ------|---------------------|---- p ∨ q | ¬p → q | p *or* q, *disjunction* p ∧ q | ¬(p → ¬q) | p *and* q, *conjunction* -p → q | ¬p ∨ q | p *implies* q, (material) *implication* (again) +p → q | ¬p ∨ q | p *implies* q, (material) *implication* p ↔ q | (p → q) ∧ (q → p) | p *if and only if* q, p *iff* q p ⊕ q | (p ∨ q) ∧ ¬(p ∧ q) | p *exclusively or* q, p *xor* q p ↑ q | ¬(p ∧ q) | p *not both* q, p *nand* q p ↓ q | ¬(p ∨ q) | *neither* p *nor* q, p *nor* q note that several of these definitions are circular. -our choice in $¬$ and $→$ as the primitive connectives is thus arbitrary. +our choice in ¬ and → as the primitive connectives is thus arbitrary. interestingly, ↑ and ↓ are *functionally complete*: we may define all other connectives in terms of them. -
+
aside: nand and nor notation | definition diff --git a/web/html.md b/web/html.md index 6e52fb2..43757b5 100644 --- a/web/html.md +++ b/web/html.md @@ -20,7 +20,7 @@ Every website you visit is composed of HTML, which alongside with styling via CS Web browsers receive HTML documents from a web server or from local storage and render the documents into multimedia web pages. HTML describes the structure of a web page semantically and originally included cues for its appearance. -HTML elements are the building blocks of HTML pages. With HTML constructs, images and other objects such as interactive forms may be embedded into the rendered page. HTML provides a means to create structured documents by denoting structural semantics for text such as headings, paragraphs, lists, links, quotes, and other items. HTML elements are delineated by tags, written using angle brackets. Tags such as and directly introduce content into the page. Other tags such as

and

surround and provide information about document text and may include sub-element tags. Browsers do not display the HTML tags but use them to interpret the content of the page. +HTML elements are the building blocks of HTML pages. With HTML constructs, images and other objects such as interactive forms may be embedded into the rendered page. HTML provides a means to create structured documents by denoting structural semantics for text such as headings, paragraphs, lists, links, quotes, and other items. HTML elements are delineated by tags, written using angle brackets. Tags such as `` and `` directly introduce content into the page. Other tags such as `

` and `

` surround and provide information about document text and may include sub-element tags. Browsers do not display the HTML tags but use them to interpret the content of the page. HTML is composed of *tags*. These tags are enclosed in angle brackets @@ -33,15 +33,15 @@ There are about XX important tags. Here they all are. ## Style -- [` `](https://developer.mozilla.org/HTML/Element/b) or [` `](https://developer.mozilla.org/HTML/Element/strong): bold text -- [` `](https://developer.mozilla.org/HTML/Element/i) or [` `](https://developer.mozilla.org/HTML/Element/em): italic text -- [` `](https://developer.mozilla.org/HTML/Element/s): struckthrough text -- [` `](https://developer.mozilla.org/HTML/Element/u): underlined text -- [` `](https://developer.mozilla.org/HTML/Element/q): quoted text -- [` `](https://developer.mozilla.org/HTML/Element/small): small text -- [` `](https://developer.mozilla.org/HTML/Element/sub): subtext -- [` `](https://developer.mozilla.org/HTML/Element/sup): supertext -- [` `](https://developer.mozilla.org/HTML/Element/code): text styled like code +- [``](https://developer.mozilla.org/HTML/Element/b) or [``](https://developer.mozilla.org/HTML/Element/strong): bold text +- [``](https://developer.mozilla.org/HTML/Element/i) or [``](https://developer.mozilla.org/HTML/Element/em): italic text +- [``](https://developer.mozilla.org/HTML/Element/s): struckthrough text +- [``](https://developer.mozilla.org/HTML/Element/u): underlined text +- [``](https://developer.mozilla.org/HTML/Element/q): quoted text +- [``](https://developer.mozilla.org/HTML/Element/small): small text +- [``](https://developer.mozilla.org/HTML/Element/sub): subtext +- [``](https://developer.mozilla.org/HTML/Element/sup): supertext +- [``](https://developer.mozilla.org/HTML/Element/code): text styled like code Some emphasis must be put: these tags are **semantic**. They are not for styling! You have a much more expressive form of styling with CSS! Given the existence of a quote key on most ASCII keyboards, this is not very useful, unless you're a stickler for Unicode quotes. @@ -76,11 +76,11 @@ These wrappers have no style associated with them, but are *semantically meaning ## Lists -- [`
    • `](https://developer.mozilla.org/HTML/Element/li): +- [`
  • `](https://developer.mozilla.org/HTML/Element/li): A *list element*. Goes within either `
      ` or `
        `. When within `
          `, `value` indicates the list numbering should skip to its value. -- [`
          `](https://developer.mozilla.org/HTML/Element/ul): +- [`
            `](https://developer.mozilla.org/HTML/Element/ul): An *unordered* list. Contains a bunch of `
          • `s. Displayed with bullet points by default. -- [`
            `](https://developer.mozilla.org/HTML/Element/ol): +- [`
              `](https://developer.mozilla.org/HTML/Element/ol): An *ordered* list. Contains a bunch of `
            1. `s. Displayed as a numbered list by default. `type` indicates the numbering type, and can be one-of `a` (lowercase ASCII), `A` (uppercase ASCII), `i` (lowercase Roman), `I` (uppercase Roman), `1` (Arabic). `start` indicates the number to start from. The presence of `reversed` indicates the list ordering should be reversed. ## Stuff -- cgit v1.2.3-70-g09d2