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.

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

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'

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

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'

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