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Foundations

Explore concepts, laws, equations and methods, with clear explanations of their use and sources. Choose an area of study or find a specific entry.

Connections

Reference entries can connect several areas of study. The same entry may appear in more than one area.

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

Showing 1–50 of 3184 relationships

Scientific relationships · 50 / 3137

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

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • The cardinality of a finite set is its number of distinct elements. The empty set has cardinality zero.
  2. Uses definition
    Relationship sources · 1

    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
    Relationship sources · 1

    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.
  4. Uses definition
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    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
    Relationship sources · 1

    A conversion changes the numerical representation of the same quantity.

  26. Uses definition
    Relationship sources · 2

    Calibration-chain uncertainty belongs to the traced measurement result.

  27. Uses definition
    Relationship sources · 1

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

  28. Related to
    Relationship sources · 1

    Propagation requires an explicit measurement model containing significant influence quantities.

  29. Derived from
    Relationship sources · 1

    The constant-mass Newtonian particle theorem follows by integrating Newton's second law along the displacement.

  30. Derived from
    Relationship sources · 1

    For a fixed classical particle system, summed Newtonian forces give the total momentum balance.

  31. Related to
    Relationship sources · 1

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

  32. Related to
    Relationship sources · 1

    The electric field defines the electric contribution to force on a test charge.

  33. Uses definition
    Relationship sources · 1

    The charge velocity and magnetic field must be evaluated in the same frame.

  34. Uses definition
    Relationship sources · 2

    The linear device relation uses current and potential difference in its stated operating range.

  35. Uses definition
    Relationship sources · 1

    Heat and work cross the specified closed-system boundary under the adopted sign convention.

  36. Uses definition
    Relationship sources · 2

    The total entropy inequality applies to the whole isolated system, not each arbitrary subsystem.

  37. Related to
    Relationship sources · 2

    The gas equation is exact within the ideal model and approximate for a real gas.

  38. Uses definition
    Relationship sources · 2

    The photon relation E = hf uses the Planck constant with energy and frequency in the same frame.

  39. Uses definition
    Relationship sources · 1

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

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

  41. Related to
    Relationship sources · 2

    Equal-and-opposite internal particle-force pairs cancel in the fixed-system Newtonian linear-momentum balance.

  42. Derived from
    Relationship sources · 1

    For a particle about a fixed inertial origin, differentiating r × p and applying Newton's second law gives net torque.

  43. Derived from
    Relationship sources · 1

    Balancing gravity and pressure forces on a static uniform-density fluid element gives the depth pressure difference.

  44. Derived from
    Relationship sources · 1

    The net hydrostatic pressure force equals displaced-fluid weight under the stated static-fluid and boundary assumptions.

  45. Uses definition
    Relationship sources · 2

    E is the electric field in the classical free-space field equations; this is not a material D/H constitutive law.

  46. Uses definition
    Relationship sources · 1

    Current density is charge flow per area and obeys local continuity with charge density in this model.

  47. Uses definition
    Relationship sources · 2

    The one-dimensional scalar-potential time-evolution equation uses ℏ = h/(2π).

  48. Related to
    Relationship sources · 1

    For the stated real-potential evolution and suitable boundaries, the normalized wavefunction's total position probability is preserved.

  49. Related to
    Relationship sources · 1

    Linked statements

    • Isotopes are atoms of the same element with different numbers of neutrons. Their atomic number is the same, while their mass number A, the total number of protons and neutrons, differs.
    • An atom has a nucleus surrounded by electrons. Its atomic number Z is the number of protons in the nucleus and identifies the element. A neutral atom has the same number of electrons as protons; gaining or losing electrons forms an ion of that element.

    Isotope identity uses the shared atomic number of an element.

  50. Related to
    Relationship sources · 1

    Linked statements

    • Stoichiometric coefficients in a balanced chemical equation specify ratios of reactant consumption and product formation. These particle-number ratios also give mole ratios, so a known reacting amount can be converted into another reacting or produced amount.
    • The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 × 10²³ specified elementary entities. The amount n and entity count N are related by n = N/N_A, where N_A = 6.02214076 × 10²³ mol⁻¹ exactly.

    Balanced particle-number ratios also give reacting mole ratios.