GLOBAL KNOWLEDGE ARK

Reference library

Foundations

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Connections

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Explore relationships within this area of study and its connections to other areas.

Showing 1–50 of 901 relationships

Scientific relationships · 50 / 867

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
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    Linked statements

    • The cardinality of a finite set is its number of distinct elements. The empty set has cardinality zero.
  2. Uses definition
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    Linked statements

    • A claim that P(x) holds for every x in a specified domain is false if an element of that domain makes P(x) false. Testing some successful examples does not establish the claim for an infinite domain.
  3. Uses definition
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    Linked statements

    • Axioms are assumptions accepted within a mathematical system. A proof derives a proposition from those assumptions and previously established results through permitted logical deductions.
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    Linked statements

    • If P(0) holds and P(n) implies P(n+1) for every nonnegative integer n, mathematical induction establishes P(n) for every nonnegative integer n.
  5. Uses definition
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    Linked statements

    • A function f:A→B assigns exactly one element of B to each element of A. A is the domain, B the codomain, and the range consists of outputs actually attained.
  6. Uses definition
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    Linked statements

    • The slope of the line through two points with different x-coordinates is (y₂ − y₁)/(x₂ − x₁). This ratio is constant along a straight line. A vertical line has no finite slope in this coordinate convention.
  7. Uses definition
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    Linked statements

    • A plotted point represents values through the graph's stated coordinate system. Read each coordinate against its own axis scale; points on the graph of an equation satisfy that equation.
  8. Uses definition
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    Linked statements

    • A planar Euclidean vector can be represented by ordered components in a chosen basis. Vector addition adds corresponding components, and scalar multiplication multiplies every component by the scalar.
  9. Uses definition
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    Linked statements

    • For a real angle θ, the point reached on the unit circle has coordinates (cos θ, sin θ). Therefore cos²θ+sin²θ=1 for every real θ.
  10. Uses definition
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    Linked statements

    • For a real angle θ, the point reached on the unit circle has coordinates (cos θ, sin θ). Therefore cos²θ+sin²θ=1 for every real θ.
  11. Uses definition
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    Linked statements

    • A complex number is written z=a+bi with real a and b and i²=−1. Its real part is a and its imaginary part is b. It is represented by (a,b) in the complex plane; addition combines the two components and multiplication uses i²=−1.
  12. Uses definition
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    Linked statements

    • A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
  13. Uses definition
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    Linked statements

    • A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
  14. Uses definition
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    Linked statements

    • The limit of f(x) as x approaches a is L if every positive output tolerance ε admits a positive input tolerance δ such that 0 < |x−a| < δ implies |f(x)−L| < ε.
  15. Uses definition
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    Linked statements

    • For a real function defined on an open interval around a, its derivative at a is the limit of [f(a+h)−f(a)]/h as h approaches zero, if that finite limit exists.
  16. Uses definition
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    Linked statements

    • For a continuous real function on [a,b], the definite integral is the limit of sums f(xᵢ*)Δx over successively finer equal subdivisions. It measures signed accumulation; negative contributions subtract rather than become positive area.
  17. Uses definition
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    Linked statements

    • If f is continuous on [a,b], the accumulation F(x)=∫ₐˣf(t)dt satisfies F′(x)=f(x) for a<x<b. For any antiderivative G of f, ∫ₐᵇf(x)dx=G(b)−G(a).
  18. Uses definition
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    Linked statements

    • If f is continuous on [a,b], the accumulation F(x)=∫ₐˣf(t)dt satisfies F′(x)=f(x) for a<x<b. For any antiderivative G of f, ∫ₐᵇf(x)dx=G(b)−G(a).
  19. Uses definition
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    Linked statements

    • A probability space consists of an outcome set Ω, a sigma-algebra of events and a probability P. Probabilities are nonnegative, P(Ω)=1, and the probability of a countable union of pairwise disjoint events equals the sum of their probabilities.
  20. Uses definition
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    Linked statements

    • For P(B)>0, the conditional probability of A given B is P(A∩B)/P(B). Events A and B are independent when P(A∩B)=P(A)P(B).
  21. Uses definition
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    Linked statements

    • For a discrete real-valued random variable X with probabilities p(x), its finite expected value is Σx xp(x) when Σx |x|p(x) is finite. The expected value need not itself be a possible observed value.
  22. Uses definition
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    Linked statements

    • For a finite discrete distribution with probabilities pᵢ, Shannon entropy is −Σᵢ pᵢ log₂ pᵢ, measured in bits. A zero-probability term contributes zero by its limiting value.
  23. Uses definition
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    Linked statements

    • There is no algorithm that always terminates and correctly decides, for every program-input pair in a Turing-complete computation model, whether that program eventually halts on that input.
  24. Uses definition
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    Linked statements

    • For nonnegative cost functions, T(n)=O(g(n)) means there are constants c>0 and n₀ such that T(n)≤cg(n) for every n≥n₀. A time or space complexity bound must specify the computation model, input-size measure and case being analysed.
  25. Uses definition
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    A conversion changes the numerical representation of the same quantity.

  26. Uses definition
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    Calibration-chain uncertainty belongs to the traced measurement result.

  27. Uses definition
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    The first-order formula combines input standard uncertainties and covariances.

  28. Related to
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    Propagation requires an explicit measurement model containing significant influence quantities.

  29. Related to
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    In the stated Lagrangian setting, continuous symmetries give conserved quantities; this is not a claim about all visual or discrete symmetries.

  30. Uses definition
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    The flat-spacetime interval uses the vacuum speed c; its sign convention is explicit.

  31. Related to
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    Decimal prefix factors supply exact scale conversions for compatible SI units.

  32. Uses definition
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    Linked statements

    • In classical first-order logic, ∀x φ holds when φ holds for every domain element; ∃x φ holds when it holds for at least one. Free variables require an assignment.
  33. Uses definition
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    Linked statements

    • ZFC combines extensionality, empty-set, pairing, union, power-set, infinity, foundation and choice axioms with separation and replacement schemas. Separation selects elements from an already given set.
  34. Uses definition
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    Linked statements

    • A set is countably infinite if it has a bijection with the natural numbers. Here, countable means finite or countably infinite, equivalently admitting an injection into N.
  35. Uses definition
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    Linked statements

    • A relation on X is an equivalence relation when it is reflexive, symmetric and transitive. Its classes [x] = {y ∈ X: y ∼ x} partition X.
  36. Uses definition
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    Linked statements

    • A first-order model interprets a theory's sentences as true. Soundness and completeness equate derivability with truth in every model. A consistent effectively axiomatized extension of Peano arithmetic nevertheless leaves some arithmetic sentence and its negation unprovable.
  37. Uses definition
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    Linked statements

    • For n ≥ 2, a ≡ b (mod n) means n divides a−b. Addition and multiplication descend to the n congruence classes; cancellation or division is not valid without invertibility.
  38. Uses definition
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    Linked statements

    • If positive moduli m,n are coprime, then any prescribed residues a,b determine exactly one residue x modulo mn satisfying x ≡ a (mod m) and x ≡ b (mod n).
  39. Uses definition
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    Linked statements

    • If positive moduli m,n are coprime, then any prescribed residues a,b determine exactly one residue x modulo mn satisfying x ≡ a (mod m) and x ≡ b (mod n).
  40. Uses definition
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    Linked statements

    • A group homomorphism φ:G→H satisfies φ(ab)=φ(a)φ(b). It preserves identity and inverses. Its kernel, the elements mapped to H's identity, is a normal subgroup of G.
  41. Uses definition
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    Linked statements

    • A ring has an additive abelian group and associative multiplication distributing over addition on both sides. Here a unital ring also has a multiplicative identity; multiplication need not commute.
  42. Uses definition
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    Linked statements

    • A field is a commutative unital ring with 0 ≠ 1 in which every nonzero element has a multiplicative inverse.
  43. Uses definition
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    Linked statements

    • Every nonconstant polynomial with complex coefficients has a complex root. Repeated factorization expresses a degree-n polynomial as a product of n linear factors times its leading coefficient, counting multiplicity.
  44. Uses definition
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    Linked statements

    • A metric d on X is nonnegative, vanishes exactly when its arguments coincide, is symmetric and satisfies the triangle inequality. Open balls contain points at distance less than a positive radius.
  45. Uses definition
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    Linked statements

    • A topology on X contains ∅ and X and is closed under arbitrary unions and finite intersections. A subset is closed when its complement in X is open.
  46. Uses definition
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    Linked statements

    • A subset K of a topological space is compact if every open cover of K has a finite subcover. For subsets of R with its usual topology, compactness is equivalent to being closed and bounded.
  47. Uses definition
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    Linked statements

    • A map f:X→Y between topological spaces is continuous exactly when f⁻¹(U) is open in X for every open subset U of Y.
  48. Uses definition
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    Linked statements

    • A vector space over a field F has an additive abelian group and scalar multiplication satisfying (ab)v=a(bv), a(u+v)=au+av, (a+b)v=av+bv and 1v=v.
  49. Uses definition
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    Linked statements

    • A basis is a linearly independent spanning set. In a finite-dimensional vector space every basis has the same number of elements, its dimension.
  50. Uses definition
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    Linked statements

    • For a linear map T:V→W with finite-dimensional V, dim V = dim ker T + dim im T. Rank is the image dimension and nullity is the kernel dimension.