diff options
Diffstat (limited to 'mathematics')
| -rw-r--r-- | mathematics/algebra.md | 75 | ||||
| -rw-r--r-- | mathematics/logic.md | 14 |
2 files changed, 82 insertions, 7 deletions
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. -<details> +<details markdown="block"> <summary>aside: nand and nor</summary> notation | definition |