Mathematical logic and foundations Definition
Ordered pair
Builds on Set
Definition
An ordered pair has a first entry and a second entry , with this property: In set theory, the Kuratowski definition is . It has the property above.
Remarks
In general , but the sets and are equal. Ordered pairs are the elements of a Cartesian product. An ordered triple can be defined as .
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Element definition
- Set definition
- Ordered pair definition
Where this leads
947 pages build on this one.
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- Partial order
- Chernoff bound
- Chapman–Kolmogorov equation
- Bias of an estimator
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- Order of cardinals
- Equivalence classes form a partition
- Triangle inequality for a metric
- Antisymmetric relation
- Infinite set
- Isolated point
- Composition of relations
- Bijective iff it has a two-sided inverse
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- Open set
- Law of the unconscious statistician
- Well-founded relation
- Left inverse
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- Exponential distribution
- Covariance matrix
- Open balls form a basis of the metric topology
- Chebyshev's inequality
- Isometries are injective
- Characteristic function of a sum of independent random variables
- Stationary distribution
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- Correlation coefficient
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- The reals have the cardinality of the continuum
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- Function composition
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- Heine–Borel theorem
- Normal distribution
- Distance from a point to a set
- Absolutely convergent series in a Banach space converge
- Equivalence class
- Mean of the Poisson distribution
- Isometry
- Statistical hypothesis test
- Binary relation
- Equinumerous sets
- Complete metric space
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- Sequence spaces lp
- Independence means the joint law is the product of the marginals
- Topological embedding
- Order of an element divides the order of the group
- Median
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- Riesz's lemma
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- Fixed points of a p-group action
- Dirac measure
- Open sets characterised by neighbourhoods
- Second Sylow theorem
- Closed sets contain their limits
- Completeness of the space of bounded continuous functions
- Method of moments
- Groups of order p squared are abelian
- Metrizable space
- Finite intersection of open sets is open
- Mean of the normal distribution
- Intersection of closed sets is closed
- The number of Sylow subgroups is the index of the normaliser
- Product of compact sets is compact
- Open sets of a metric space
- Metric topology
- Baire category theorem
- Schröder–Bernstein theorem
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- Metrizable spaces are normal
- Normal space
- Independent and identically distributed random variables
- Classical definition of probability
- Marginal distribution
- Moment-generating function of the normal distribution
- Open balls are open
- Quantile function
- Continuity via closed sets
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- Finite set
- Banach space
- Weak law of large numbers
- Random walk
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- Random variable with a density
- Interior
- Uncorrelated random variables that are not independent
- Cartesian product
- Kurtosis
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- Dense set
- Restriction of a function
- A set is closed iff it equals its closure
- Baire space
- Variance of a sum
- Limits are unique in Hausdorff spaces
- Connected space
- Mean of the geometric distribution
- Open ball
- Expected value
- Right inverse
- Strong law of large numbers
- Covariance
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- Closed subsets of complete spaces are complete
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- Epsilon-delta uniform continuity
- Lévy's continuity theorem
- Convergence of random variables
- Meagre set
- Maximum likelihood estimation
- Identity function
- Completion of a metric space
- Multivariate normal distribution
- Closed set
- Lexicographic order
- Separable space
- The alternating group A5 is simple
- Antichain
- Chain
- Cardinality of the power set
- Skewness
- Pseudometric space
- Upper bound
- Galton–Watson process
- Interior is the largest open subset
- Markov property
- Limit point
- Union of open sets is open
- First-countable space
- Axiom of choice
- Injective iff it has a left inverse
- Student's t-test
- Equivalent norms
- Associativity of composition
- Discrete random variable
- Bernoulli distribution
- Converse relation
- Convergence in distribution
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- Regular space
- Boundary
- Abelian simple groups have prime order
- Cauchy's theorem
- Sample variance
- Cardinality
- Tychonoff's theorem
- Bayesian inference
- Path-connected space
- Topologist's sine curve
- Confidence interval
- Complement of an open set is closed
- Compactness and sequential compactness agree in metric spaces
- Continuous image of a compact set is compact
- Hausdorff maximal principle
- Nowhere dense set
- Variance under scaling
- Discrete uniform distribution
- The number of Sylow subgroups divides the index
- Universal property of the product topology
- Epsilon characterisation of Cauchy sequences
- Minimal element
- One-point compactification
- Continuum hypothesis
- Compact Hausdorff spaces are normal
- Contraction mapping
- Equicontinuity
- Hausdorff space
- Coupon collector's problem
- Cumulative distribution function
- Open connected subsets of R^n are path-connected
- Characteristic function
- Compact subsets of a Hausdorff space are closed
- Well-ordering theorem
- Completeness via Cauchy sequences
- Hilbert's hotel
- Symmetric relation
- Cauchy distribution
- Variance as the mean of the square minus the square of the mean
- Epsilon-net characterisation of total boundedness
- Birthday problem
- Moment-generating function
- Cantor pairing function
- Student's t-distribution
- Connected components are closed
- Characteristic function of the normal distribution
- Open cover
- Sum of independent normal random variables
- Closed map
- Compactness via open covers
- Linearity of expectation
- Bounded set
- Metric spaces are Hausdorff
- Sup norm on bounded continuous functions
- Converse of Lagrange's theorem fails for the alternating group A4
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- Moment
- Heine–Borel theorem in finite-dimensional normed spaces
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- Interior is open
- Beta distribution
- Expectation with respect to the binomial distribution
- Injective function
- Cumulant generating function
- Image of a set
- Third Sylow theorem
- Joint probability distribution
- Equivalence relation induced by a function
- Stochastic process
- Cardinal number
- Sample mean
- Compact space
- Frattini argument
- Connected set
- T0 space
- Type I and type II errors
- Compact subsets of a metric space are bounded
- Ordinal number
- Generating function of a sum of independent random variables
- Chi-squared distribution
- Limit of the cumulative distribution function at infinity
- Cantor set is homeomorphic to the space of binary sequences
- Every infinite set has a countably infinite subset
- Image of an intersection
- Preorder
- Size of an orbit times the size of the stabiliser
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- Path component
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- Closure is the smallest closed superset
- Bijective function
- Hypergeometric distribution
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- Almost sure convergence implies convergence in probability
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- Bertrand paradox
- Least element
- Subspace metric
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- Lipschitz map
- Law of a random variable
- Convergence in mean
- Estimator
- Equivalent metrics
- Surjective iff it has a right inverse
- Maximal element
- Variance of a sum of independent random variables
- Aleph-null
- Continuous real function on a compact set attains its bounds
- Mean squared error
- Quotient topology
- First Sylow theorem
- Poisson distribution
- Continuity is equivalent to sequential continuity in metric spaces
- Banach fixed point theorem
- Isometric isomorphism
- Binomial distribution
- Indicator function
- Standard deviation
- Compact closed unit ball implies finite dimension
- Complete subspaces are closed
- Inverse function
- Cardinality of the continuum
- Totally bounded set
- Probability mass function
- Monotone function
- Gambler's ruin
- Continuous uniform distribution
- Circle as a quotient of the real line
- Hoeffding's inequality
- Moment-generating function of a sum of independent random variables
- Cantor's theorem
- Hölder continuity
- Countable intersection of dense open sets is dense
- Tietze extension theorem
- Uncountable set
- Independence of random variables
- Cantor's diagonal argument
- Geometric distribution
- Monotonicity of the cumulative distribution function
- Reflexive relation
- Function
- Consistent estimator
- Surjective function
- Poisson limit theorem
- Negative binomial distribution
- Graph of a function
- Tail sum formula for expectation
- Order isomorphism
- Conditional expectation
- Absorbing Markov chain
- Comparability of cardinals
- Groups of order pq
- Continuity via open sets
- Sum of independent Poisson random variables
- Closed balls are closed
- Conditional probability distribution
- Choice function
- Burnside's lemma
- Proper metric space
- Product topology
- Greatest element
- Countable union of countable sets is countable
- Probability-generating function
- Memorylessness
- g to the order of the group is the identity
- Countable set
- Transfinite induction
- Path-connected spaces are connected
- Central moment
- Operator norm
- Law of large numbers
- Quotient set
- Doob's upcrossing inequality
- Relative CW complex
- Proof complexity
- Riemann domain over C^n
- Closure operator
- Univalent function
- Translation surface
- Bernstein–von Mises theorem
- Busemann G-space
- Arnold tongue
- Cramér–Rao bound
- Linearly ordered group
- Stone–Weierstrass theorem
- Pettis measurability theorem
- Efficiency of an estimator
- Log-rank test
- Arakelyan's theorem
- Kirkman's schoolgirl problem
- Functional data analysis
- Weak ordering
- Poisson regression
- Lebesgue's theorem on differentiability of monotone functions
- Sequential analysis
- Locally compact group
- Ergodic decomposition
- Nonexistence of infinite-dimensional Lebesgue measure
- Expansion of iterated differences in shifts
- Free monoid
- Order type
- Ordinal arithmetic
- Discrete series representation
- Kleene algebra
- Köthe–Toeplitz dual
- Koksma–Hlawka inequality
- Hodges' superefficient estimator
- Martingale representation theorem
- Matrix Wiener–Hopf factorization
- Backward difference operator
- Rectifiable set
- Repair radii and villainy
- Classification of surfaces
- Namioka--Phelps
- Mathematical statistics
- F-test
- Laplace transform of the Dirac delta
- Stationary process
- Twelvefold way
- Lagrange's theorem
- Nilradical is the intersection of the prime ideals
- L2 convergence of Fourier series
- Cramér–Wold theorem
- Oscillator representation
- Ultrafilter lemma
- Brouwer's characterization of the Cantor space
- Boolean model
- Convergence of the Neumann series for Volterra kernels
- Exponential of a bounded operator
- Von Neumann universe
- Kolmogorov–Arnold–Moser theorem
- Schwarz–Christoffel mapping
- Type theory
- Stochastic partial differential equation
- Latin square
- Series in Banach spaces
- Teichmüller space
- Galois group of a prime degree polynomial with two nonreal roots
- Minkowski question-mark function
- Wald–Wolfowitz optimality of the sequential probability ratio test
- Doob's martingale inequality
- Sequential probability ratio test
- Céa's lemma
- Brownian bridge
- Consistency of the maximum likelihood estimator
- Wiener process
- Fodor's lemma
- Empirical Bayes method
- Ordinal analysis
- Woodin cardinal
- Karhunen–Loève theorem
- Pettis integral
- Strongly measurable function
- Brent's method
- Mergelyan's theorem
- Itô isometry
- Naive set theory
- Matrix Riemann–Hilbert problem
- CW complex
- Indefinite sum
- High indices theorem
- Hille–Yosida theorem
- Bayes factor
- Diaconescu's theorem
- Pickands–Balkema–De Haan theorem
- Global attractor
- Skolem's paradox
- Runge's theorem
- Stationary set
- Basu's theorem
- Cantor normal form
- Bump function
- Vanishing mean oscillation
- Spectral decomposition theorem
- Banach bundle
- Veblen function
- Axiom of regularity
- Variogram
- Robbins–Monro algorithm
- Contiguity of measures
- Order statistic
- Besicovitch covering theorem
- Riesz space
- Dirichlet process
- Absolute neighborhood retract
- Lindley's paradox
- Tychonoff space
- Model theory of algebraically closed fields
- M-estimator
- Fixed-point iteration
- Kunen's inconsistency theorem
- Ornstein–Uhlenbeck process
- Kernel regression
- Lp space
- Hilbert's twenty-first problem
- Bisection method
- Interior algebra
- Adapted process
- Score test
- Kingman's subadditive ergodic theorem
- Multivariate analysis of variance
- Nichols–Zoeller theorem
- Lax–Milgram theorem
- McDiarmid's inequality
- Lévy's characterization of Brownian motion
- Asymptotic normality of the maximum likelihood estimator
- Mann–Whitney U test
- Central difference
- Neumann series
- Kolmogorov automorphism
- Singular cardinals hypothesis
- Standard map
- Score
- Hall's theorem on solvable groups
- F-distribution
- Shift operator
- Δ-complex
- Difference quotient
- Lebesgue covering dimension
- Axiom schema of replacement
- Edgeworth expansion
- Bourbaki–Witt theorem
- Internal set theory
- Continuous functional calculus
- Summation by parts
- Hausdorff paradox
- Monotone likelihood ratio
- Δ-system lemma
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Brelot space
- Symmetric derivative
- Nash–Moser theorem
- Metropolis–Hastings algorithm
- Partially ordered ring
- Ordered group
- Minimum total potential energy principle
- Continuous-time Markov chain
- Darboux function
- Modulus of continuity
- Banach limit
- Kriging
- Cantor's isomorphism theorem
- Identifiability
- Baire function
- Branched covering
- Minimal sufficient statistic
- Order dimension
- Whittle likelihood
- Logistic regression
- Carathéodory's theorem on boundary extension
- Order embedding
- Convergence to equilibrium of ergodic Markov chains
- Cramér–von Mises criterion
- Azuma's inequality
- Fréchet manifold
- Gaussian free field
- Parry measure
- Kernel density estimation
- Hardy–Littlewood maximal inequality
- Proof that the order of an element divides the order of the group
- Interior operator
- Duality in order theory
- Arnold's cat map
- Landau's constants
- Implicit function theorem in Banach spaces
- Upper set
- Erdős–Kac theorem
- Proof-theoretic ordinal
- Random group
- Ordinal analysis
- Proof-theoretic ordinal
- Archimedean equivalence classes
- Detailed balance
- Strict total order
- Coupling from the past
- Steinitz's theorem on algebraically closed fields
- Robust statistics
- Lebesgue–Stieltjes integration
- Lipschitz domain
- Hölder's theorem on Archimedean groups
- Hidden Markov model
- Finite-dimensional distributions
- Continuous mapping theorem
- Bias–variance decomposition
- Category of posets
- Lévy process
- Wald's equation
- Random ordinary differential equation
- Topological ring
- Finite difference
- Birkhoff's subdirect representation theorem
- Mahler's theorem
- Suslin's problem
- Hadamard gap theorem
- Monte Carlo integration
- Time series
- Karamata's Tauberian theorem
- Optional stopping theorem
- Newton's forward difference formula
- Integral operator
- Singular simplicial set
- Breakdown point
- Delta method
- Transformation semigroup
- Regular surface
- Sokhotski–Plemelj formulae
- Newton series
- Hall subgroup
- Higher-order differences
- Noncentral chi-squared distribution
- Characterizations of semisimple modules
- Hájek–Le Cam convolution theorem
- Repair radii and villainy
- Differences of polynomials vanish beyond the degree
- Harmonic measure
- Covering relation
- Behrens–Fisher problem
- FK-space
- Cohen's independence of CH
- Anosov flow
- Bochner integral
- Submartingale
- Hodges–Lehmann estimator
- Kruskal–Wallis test
- Midpoint convex function
- Stochastic difference equation
- f-ring
- Sampling distribution
- Vitali covering lemma
- Sylow theorems
- Information geometry
- Discontinuous additive functions via a Hamel basis
- Banach–Tarski paradox
- Recurrence and transience of states
- Approximation of a kernel by degenerate kernels
- Complex p-adic numbers
- Continuous function with divergent Fourier series
- Malliavin calculus
- Wishart distribution
- Birth–death process
- Hahn embedding theorem
- de Rham curve
- Statistical inference
- Fredholm map
- Minimum-variance unbiased estimator
- Frattini subgroup
- Sign test
- Bounded sequences ℓ∞
- Lattice-ordered group
- Sequence space
- Proof that groups of prime order are cyclic
- Krasner's lemma
- Action-angle coordinates
- Slutsky's theorem
- Tanaka's formula
- Reduction of a first-kind Volterra equation to the second kind
- Several complex variables
- Space of convergent sequences
- Club set
- Sinai billiard
- Liouville–Neumann series
- Geometric Brownian motion
- Empirical likelihood
- Dvoretzky–Kiefer–Wolfowitz inequality
- Bayesian information criterion
- Dvoretzky–Rogers theorem
- Berry–Esseen theorem
- Hilbert metric
- Hardy–Littlewood–Sobolev inequality
- Hunt process
- Cochran's theorem
- Stochastic approximation
- Kronecker's approximation theorem
- Jackknife
- Quantile regression
- Markov chain mixing time
- Birkhoff's ergodic theorem
- Diamond principle
- Dense order
- Homotopy type theory
- Akaike information criterion
- Schauder fixed-point theorem for integral equations
- Ordered ring
- Milnor's isospectral flat tori
- Inverse transform
- Set theory
- Poincaré inequality
- Hewitt–Savage zero–one law
- Berkovich space dynamics
- Reflection principle
- Credible interval
- Conformal modulus
- Combinatorial species
- Contraction mapping principle for integral equations
- Intuitionistic type theory
- Density of trigonometric polynomials in continuous functions
- Jordan decomposition of a function of bounded variation
- Completion of a valued field
- Orbifold
- Mutually orthogonal Latin squares
- Ill-posedness of first-kind equations
- Proper forcing axiom
- Aronszajn tree
- Nonexistence of a projective plane of order 10
- Wiener–Ikehara theorem
- Independence of sample mean and sample variance for normal samples
- Self-similar process
- Gibbs sampling
- Epsilon nought
- Estimating equations
- Bloch's theorem
- Spectral radius formula
- Supermartingale
- Anderson–Darling test
- Spearman's rank correlation coefficient
- Inverse function theorem in Banach spaces
- Family-wise error rate
- Subshift of finite type
- Backward Euler method
- Fisher's inequality
- Galois connection
- Torus knot
- Markov chain Monte Carlo
- Suslin line
- Nested intervals theorem
- Compactness of integral operators
- Hankel operator
- Dickey–Fuller test
- Orientation of a manifold
- Fundamental theorem of the calculus of finite differences
- Galerkin method
- Baer's criterion
- Generalized method of moments
- Doob's martingale convergence theorems
- Random integral equation
- Uniformly most powerful test
- Compactness of families of harmonic functions
- Real projective plane
- Bernstein inequalities
- Kendall rank correlation coefficient
- Interchange of summation order
- Semimartingale
- Empirical distribution function
- Bootstrap
- Curtis–Hedlund–Lyndon theorem
- Lawvere's fixed-point theorem
- Idempotent semiring
- Influence function
- Orlicz–Pettis theorem
- Oseledets theorem
- Itô calculus
- Alexander subbase theorem
- Cantor function
- Ancillary statistic
- Commutative Gelfand–Naimark theorem
- Cofinality
- Stopped process
- Tauberian theorems for integral transforms
- Transcendence degree
- Enumerative combinatorics
- Fourier basis of L2 of the circle
- Hyperbolic set
- Harnack's principle
- Moments of the log-normal distribution
- Hilbert manifold
- Isotonic regression
- Jentzsch's theorem on positive kernels
- Complete statistic
- Horvitz–Thompson estimator
- Nörlund–Rice integral
- Quasisymmetric map
- Liouville–Arnold theorem
- U-statistic
- Fractional Brownian motion
- L-estimator
- Doubling on the integers is not an isomorphism
- Normal function
- Power of a test
- Integral transform
- Banach–Stone theorem
- Gödel's consistency of CH and AC
- Banach–Mazur theorem
- Steiner system
- Left-orderable group
- Moore–Osgood theorem
- Sufficient statistic
- Gentzen's consistency proof
- Gaussian process
- Categorical theory
- Normal family
- Missing data mechanisms
- Bruck–Ryser–Chowla theorem
- Ordered semigroup
- Likelihood-ratio test
- Cardinal characteristics of the continuum
- Karlin–Rubin theorem
- Sokhotski–Plemelj theorem
- Noncommutative geometry
- Riemann–Hilbert problem
- Schauder basis
- Erdős–Rado theorem
- Axiom A
- Tsirelson space
- Conjugate prior
- Kolmogorov–Smirnov test
- Lebesgue differentiation theorem
- F-space
- Ladyzhenskaya–Babuška–Brezzi condition
- Maximum and minimum
- Singular homology
- Delta derivative on a time scale
- Expectation–maximization algorithm
- Carleman's singular integral equation
- Ultrafilter
- Hotelling's T-squared distribution
- Boolean prime ideal theorem
- Strict partial order
- Burkholder–Davis–Gundy inequality
- Doob–Meyer decomposition theorem
- Resolvent as Laplace transform of a semigroup
- Cross-validation
- Bounded poset
- Residuated mapping
- Wald test
- Szpilrajn extension theorem
- Extremal length
- Aleph number
- Pivotal quantity
- Banach algebra
- Down-closure
- Block design
- Generalized linear model
- Hahn series
- Iterated function system
- Axiom of dependent choice
- Tame Fréchet space
- Forward difference operator
- Von Neumann–Bernays–Gödel set theory
- Basel problem via Fourier series
- Box–Jenkins method
- Kolmogorov continuity theorem
- Banach–Lie group
- Krull topology
- Banach manifold
- BK-space
- Graph embedding
- Pesin theory
- Thom–Mather stratified space
- Cantor algebra
- Empirical process
- Convenient vector space
- Binomial proportion confidence interval
- Pseudoanalytic function
- Lévy's upward theorem
- Proof of Lagrange's theorem
- Class
- Existence of transcendence bases
- Composition operator
- False discovery rate and the Benjamini–Hochberg procedure
- Cantor space
- Neyman construction
- Axiom of determinacy
- Space of null sequences
- Natural boundary
- Riemann surface
- Unique solvability of Volterra equations of the second kind
- Positive cone
- Easton's theorem
- Wilks' theorem
- Fubini's theorem for series
- Euler's thirty-six officers problem
- Vitali set
- Maximal chain
- Pseudo-Anosov map
- Error-correcting code
- Anosov diffeomorphism
- Analytic capacity
- Ordered vector space
- Mittag-Leffler's theorem
- Algorithmic randomness
- Harmonic space
- Fubini's theorem on differentiation
- Stability conjecture
- Geometric realization
- Skorokhod's embedding theorem
- Galois insertion
- Prime end
- Torus as a group
- Function of exponential order
- Semi-continuity
- Infinite compositions of functions
- Permutation test
- Fisher's exact test
- Bonferroni correction
- Automorphism group of a bounded domain is a Lie group
- Doob decomposition theorem
- Rellich–Kondrachov theorem
- Local asymptotic normality
- Oscillation of a function
- Frequentist inference
- Likelihood principle
- Vaught conjecture
- Bunimovich stadium
- Maximal torus
- Summation of families over arbitrary index sets
- Fisher–Neyman factorization theorem
- Girsanov theorem
- Symmetric and self-reciprocal kernels
- Stone–von Neumann theorem
- Martin's axiom
- Cauchy's functional equation
- Quadratic variation
- Beth number
- Regular cardinal
- Faber polynomials
- Observed information
- Feller process
- Point estimation
- Montel's theorem
- Local martingale
- Orderable group
Further reading
- Web https://en.wikipedia.org/wiki/Ordered_pair, lead section
About this page
| Tag | 0JQWWFCS9R |
|---|---|
| URI | m2://0JQWWFCS9R |
| Kind | definition |
| Writing | W1 stated |
| MSC | 03E20 |
| Defines | Ordered pair |
| Uses | Set |
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