# outline of mathematics
# five important generalizations in mathematics
* group theory generalizes algebraic operations by abstracting the notion of symmetry and invertible structure.
* topology generalizes geometry by focusing on continuity and qualitative spatial properties rather than metric or angle.
* measure theory generalizes length, area, and volume in a rigorous way, enabling integration beyond riemann sums.
* category theory generalizes mathematical structures and functions via objects and morphisms, unifying various branches.
* functional analysis generalizes linear algebra and calculus to infinite-dimensional vector spaces.

# foundations
- logic:
  - propositions: statements with definite truth value
  - connectives: and, or, not, implies, iff
  - quantifiers: for all, there exists
  - inference rules: modus ponens, modus tollens, etc.
  - proof systems:
    - natural deduction
    - sequent calculus
    - Hilbert-style systems
    - formal vs informal proofs
- axioms:
  - definition: statements taken as true without proof, forming a base for reasoning
  - axiom schemas: patterns generating infinitely many axioms
  - example axiom systems:
    - Zermelo–Fraenkel set theory with Choice (ZFC)
    - Peano axioms for arithmetic
    - Euclidean geometry postulates
    - Group axioms
  - independence and consistency of axioms
- proofs:
  - direct proof: deduce conclusion from premises via valid steps
  - proof by contradiction: assume negation, derive a contradiction
  - contrapositive proof: prove "if not q then not p" instead of "if p then q"
  - proof by induction:
    - weak induction
    - strong induction
    - structural induction
  - constructive proof: explicitly produce an example
  - non-constructive proof: show existence without explicit construction
  - exhaustion (case analysis)
  - probabilistic proofs
  - computer-assisted proofs
- definitions:
  - formal definitions: precise specification of objects or properties
  - recursive definitions: define in terms of smaller instances
  - implicit definitions: specify properties uniquely determining an object
  - constructive vs non-constructive definitions
- notation:
  - symbolic notation: algebraic, logical, set-theoretic symbols
  - syntactic conventions: operator precedence, associativity, bracketing
  - semantic conventions: meaning of symbols in context
  - index and summation notation
  - functional notation: f(x), λx, etc.
# objects and structures
- sets and collections:
  - finite and infinite sets
  - subsets and supersets
  - power sets
  - Cartesian products
  - indexed families of sets
- numbers and algebraic systems:
  - natural numbers
  - integers
  - rational numbers
  - real numbers
  - complex numbers
  - quaternions and other extensions
  - modular arithmetic
- functions and relations:
  - total and partial functions
  - relations and their properties (reflexive, symmetric, transitive)
  - equivalence relations and partitions
  - order relations: partial, total, well-orderings
- sequences, series, limits:
  - arithmetic and geometric sequences
  - convergence and divergence
  - infinite series and sums
  - limit definitions
- algebraic structures:
  - groups
  - rings
  - fields
  - modules
  - vector spaces
  - algebras over fields
- order and lattice structures:
  - partially ordered sets (posets)
  - lattices and complete lattices
  - Boolean algebras
- topological spaces:
  - open and closed sets
  - neighborhoods
  - continuity in topology
  - compactness and connectedness
- geometric and analytic structures:
  - Euclidean spaces
  - metric spaces
  - manifolds
  - measure spaces
  - Hilbert and Banach spaces
# statements and relations
- expressions: represent values; variables, constants, operators; no equals sign
  - algebraic:
    - monomial: one term (eg '5 x')
    - binomial: two terms (eg 'x ** 2 + 2 x')
    - polynomial: multiple terms (eg '4 x ** 3 - 3 x ** 2 + 2 x - 1')
  - rational: ratios of polynomials (eg '(x ** 2 - 1) / (x + 1)')
  - exponential: involving exponents (eg '2 ** x')
  - logarithmic: involving logarithms (eg 'log(x)')
  - trigonometric: involving trig functions (eg 'sin(x) + cos(x)')
  - radical: involving roots (eg 'sqrt(x + 1)')
- equations: assert equality between expressions
  - linear: first degree (eg '2 x + 3 = 7')
  - quadratic: second degree (eg 'a x ** 2 + b x + c = 0')
  - polynomial: higher degree (eg 'x ** 3 - 4 x ** 2 + x + 6 = 0')
  - rational: involving rational expressions (eg '1 / (x + 1) = 2')
  - exponential: involving exponential expressions (eg '2 ** x = 8')
  - logarithmic: involving logarithms (eg 'log(x) = 3')
  - trigonometric: involving trig functions (eg 'sin(x) = 0.5')
  - differential: involving derivatives (eg '(d ** 2 * y) / (d x ** 2) + 3 * ((d y) / (d x)) + 2 y = 0')
- inequalities: compare expressions without requiring equality
  - linear: '2 x + 3 > 7'
  - polynomial: 'x ** 2 - 4 < 0'
  - rational: '(x - 1) / (x + 2) >= 0'
  - exponential: '2 ** x < 5'
  - logarithmic: 'log(x) > 1'
  - trigonometric: 'sin(x) <= 0.5'
# operations and transformations
- algebraic operations:
  - addition, subtraction, multiplication, division
  - exponentiation and roots
- transformations:
  - linear transformations
  - affine transformations
  - nonlinear transformations
  - geometric transformations: translation, rotation, reflection, scaling
- composition and inversion:
  - function composition
  - inverse transformations
- operators:
  - differential operators
  - integral operators
  - linear operators in functional analysis
  - projection operators
  - adjoint operators
- morphisms:
  - homomorphisms
  - isomorphisms
  - endomorphisms
  - automorphisms
# properties and characteristics
- functions: map each input to exactly one output
  - types:
    - linear: 'f(x) = m x + b'
    - quadratic: 'f(x) = a x ** 2 + b x + c'
    - polynomial: 'f(x) = x ** 3 - 4 x ** 2 + x + 6'
    - rational: 'f(x) = (x ** 2 - 1) / (x + 1)'
    - exponential: 'f(x) = 2 ** x'
    - logarithmic: 'f(x) = log(x)'
    - trigonometric: 'f(x) = sin(x) + cos(x)'
    - piecewise: different expressions for different intervals
      - example: 'f(x) = x' if 'x >= 0', 'f(x) = -x' if 'x < 0'
    - implicit: defined by relation not solved for dependent variable
      - example: 'x ** 2 + y ** 2 = 1'
  - characteristics:
    - arity:
      - unary: one input (eg 'f(x) = x ** 2')
      - binary: two inputs (eg 'f(x, y) = x + y')
      - n-ary: n inputs (eg 'f(x_1, ..., x_n) = x_1 + ... + x_n')
    - domain: set of all possible inputs
      - example: for 'f(x) = sqrt(x)', domain is 'x >= 0'
    - range: set of all possible outputs
      - example: for 'f(x) = x ** 2', range is 'y >= 0'
    - codomain: set of potential outputs
      - example: for 'f: real_numbers -> real_numbers', codomain is 'real_numbers'
    - injectivity: different inputs → different outputs
      - example: 'f(x) = 2 x + 1'
    - surjectivity: every codomain element has some preimage
      - example: 'f(x) = x ** 3'
    - bijectivity: injective and surjective
      - example: 'f(x) = x + 5'
    - continuity: small input changes → small output changes
      - example: 'f(x) = x ** 2'
    - differentiability: has derivative everywhere in domain
      - example: 'f(x) = x ** 3'
    - periodicity: ∃ t > 0 s.t. 'f(x + t) = f(x)'
      - example: 'f(x) = sin(x)'
    - symmetry:
      - even: 'f(-x) = f(x)'
        - example: 'f(x) = x ** 2'
      - odd: 'f(-x) = -f(x)'
        - example: 'f(x) = x ** 3'
    - monotonicity: non-increasing or non-decreasing
      - monotonically increasing: 'f(x) = x'
      - monotonically decreasing: 'f(x) = -x'
    - boundedness: ∃ m s.t. '|f(x)| <= m'
      - bounded above: 'f(x) = 1 / (1 + x ** 2)'
      - bounded below: 'f(x) = x ** 2'
    - inverse: ∃ 'f ** -1' s.t. 'f((f ** -1)(y)) = y' and '(f ** -1)(f(x)) = x'
      - example: inverse of 'f(x) = 2 x' is '(f ** -1)(x) = x / 2'

# domains of mathematics
- algebra:
  - elementary algebra
  - linear algebra
  - abstract algebra
  - universal algebra
- analysis:
  - real analysis
  - complex analysis
  - functional analysis
  - harmonic analysis
- geometry:
  - Euclidean geometry
  - differential geometry
  - algebraic geometry
  - projective geometry
- topology:
  - general topology
  - algebraic topology
  - differential topology
- number theory:
  - elementary number theory
  - analytic number theory
  - algebraic number theory
  - computational number theory
- probability and statistics:
  - probability theory
  - descriptive statistics
  - inferential statistics
  - stochastic processes
- discrete mathematics:
  - combinatorics
  - graph theory
  - discrete structures and algorithms
  - coding theory
- mathematical logic:
  - propositional logic
  - first-order logic
  - model theory
  - proof theory
  - computability theory
  - set theory
- applied mathematics:
  - numerical analysis
  - optimization
  - operations research
  - mathematical physics
  - control theory
  - mathematical biology
  - financial mathematics
- category theory:
  - basic category theory
  - functors and natural transformations
  - limits and colimits
  - adjunctions
  - monoidal categories
  - higher category theory