Mathematical logic and foundations Definition
Union
Builds on SetElementDisjunction
Definition
Let and be sets. The union of and is the set So an object is an element of when or , or both.
Remarks
Example: . Union is commutative and associative, and . The union of an arbitrary family of sets is in union of a family of sets.
Reading path
To understand this page from the ground up, read these 5 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Element definition
- Proposition concept
- Set definition
- Disjunction definition
- Union definition
Where this leads
745 pages build on this one.
- Closure is the set of limits of sequences
- Variance
- Finite intersection property
- Prior probability
- Compact set
- Second-countable space
- Event
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Chernoff bound
- Chapman–Kolmogorov equation
- Bias of an estimator
- Totally disconnected space
- Urysohn's lemma
- Equivalence classes form a partition
- Isolated point
- Cantor set is closed
- Identically distributed random variables
- Open set
- Law of the unconscious statistician
- Density of the normal distribution
- Exponential distribution
- Covariance matrix
- Open balls form a basis of the metric topology
- Chebyshev's inequality
- Characteristic function of a sum of independent random variables
- Stationary distribution
- Closure is closed
- Bernoulli process
- Markov chain
- Statistic
- Arzelà–Ascoli theorem
- Law of total variance
- Variance of the normal distribution
- Equivalence of norms in finite dimension
- Torus
- Cauchy–Schwarz inequality for expectations
- Lebesgue number lemma
- Continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Monotonicity of probability
- Möbius strip
- Correlation coefficient
- Variance under translation
- Almost sure convergence
- Law of total expectation
- Zorn's lemma
- Product topology on an arbitrary product
- Sigma-algebra
- Probability density function
- Compact iff complete and totally bounded
- Expectation of a product of independent random variables
- Likelihood function
- Union of a family of sets
- T1 space
- Homeomorphism
- Groups of prime order are cyclic
- Composition of continuous maps is continuous
- Markov's inequality
- Standard score
- Poisson point process
- Gamma distribution
- Inclusion of a subspace is continuous
- De Moivre–Laplace theorem
- Closed subsets of compact sets are compact
- Closure in a metric space
- Heine–Cantor theorem
- Basis of a topology
- Buffon's needle problem
- Kolmogorov's zero–one law
- Correspondence between equivalence relations and partitions
- Independent random variables are uncorrelated
- Convolution of probability distributions
- Continuity at a point of a topological space
- Heine–Borel theorem
- Normal distribution
- Absolutely convergent series in a Banach space converge
- Mean of the Poisson distribution
- Statistical hypothesis test
- Distributive laws for sets
- 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
- Comparison of topologies
- Geometric probability
- Riesz's lemma
- Intermediate value theorem for connected spaces
- Fixed points of a p-group action
- Dirac measure
- Partition of a set
- 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
- Independent sigma-algebras
- 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
- Finite-dimensional normed spaces are complete
- Metrizable spaces are normal
- Normal space
- Independent and identically distributed random variables
- Classical definition of probability
- Pairwise independent events that are not mutually independent
- Marginal distribution
- Moment-generating function of the normal distribution
- Open balls are open
- Quantile function
- Continuity via closed sets
- Connected subsets of the real line are intervals
- Banach space
- Weak law of large numbers
- Multiplication rule
- Borel sigma-algebra
- Addition rule for two events
- Random walk
- Linear maps on finite-dimensional spaces are continuous
- Random variable with a density
- Interior
- Uncorrelated random variables that are not independent
- Sequential compactness
- Kurtosis
- Martingale
- Subbasis
- Topological space
- Box topology
- Dense set
- A set is closed iff it equals its closure
- Baire space
- Subspace topology
- Variance of a sum
- Limits are unique in Hausdorff spaces
- Connected space
- Probability axioms
- Mean of the geometric distribution
- Expected value
- Strong law of large numbers
- Covariance
- Random variable
- Closure
- Base rate fallacy
- Law of total probability
- Independence of two random variables
- Neyman–Pearson lemma
- Convergence in probability
- Boole's inequality
- P-value
- Epsilon-delta continuity in metric spaces
- Chi-squared test
- Continuous image of a connected set is connected
- Convergence in probability implies convergence in distribution
- Central limit theorem
- Locally compact space
- 68–95–99.7 rule
- Cantor's intersection theorem
- Transition matrix
- Conditional variance
- Bounded linear maps are continuous
- Closed subsets of complete spaces are complete
- Universal property of the quotient topology
- Conditional probability
- Lévy's continuity theorem
- Convergence of random variables
- Meagre set
- Maximum likelihood estimation
- Completion of a metric space
- Multivariate normal distribution
- Closed set
- Mutual independence of events
- Separable space
- The alternating group A5 is simple
- Skewness
- 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
- Student's t-test
- Equivalent norms
- Discrete random variable
- Bernoulli distribution
- Convergence in distribution
- The cumulative distribution function determines the distribution
- Regular space
- Boundary
- Abelian simple groups have prime order
- Cauchy's theorem
- Sample variance
- 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
- Venn diagram
- Posterior probability
- 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
- One-point compactification
- Compact Hausdorff spaces are normal
- 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
- Pairwise independence
- Cauchy distribution
- Bayes' theorem
- Variance as the mean of the square minus the square of the mean
- Birthday problem
- Moment-generating function
- Student's t-distribution
- Connected components are closed
- Characteristic function of the normal distribution
- Open cover
- Second Borel–Cantelli lemma
- Sum of independent normal random variables
- Closed map
- Independence of two events
- Compactness via open covers
- Linearity of expectation
- Metric spaces are Hausdorff
- Sup norm on bounded continuous functions
- Monty Hall problem
- Converse of Lagrange's theorem fails for the alternating group A4
- Moment
- Heine–Borel theorem in finite-dimensional normed spaces
- Projections are continuous
- Interior is open
- Beta distribution
- Probability of the complement
- Expectation with respect to the binomial distribution
- Cumulant generating function
- Third Sylow theorem
- Joint probability distribution
- Probability distribution
- Stochastic process
- Sample mean
- Compact space
- Frattini argument
- Connected set
- T0 space
- Type I and type II errors
- Almost surely
- Compact subsets of a metric space are bounded
- 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
- Size of an orbit times the size of the stabiliser
- Continuous linear map
- Path component
- Closure is the smallest closed superset
- Hypergeometric distribution
- Open map
- Moments are derivatives of the moment-generating function at zero
- Almost sure convergence implies convergence in probability
- Continuous map of topological spaces
- Bertrand paradox
- First Borel–Cantelli lemma
- The characteristic function determines the distribution
- Law of a random variable
- Convergence in mean
- Estimator
- Equivalent metrics
- Surjective iff it has a right inverse
- Variance of a sum of independent random variables
- Continuous real function on a compact set attains its bounds
- Mean squared error
- Quotient topology
- First Sylow theorem
- Poisson distribution
- Probability measure
- Continuity is equivalent to sequential continuity in metric spaces
- De Morgan's laws for sets
- Binomial distribution
- Standard deviation
- Compact closed unit ball implies finite dimension
- Complete subspaces are closed
- Law of total probability for an event and its complement
- Probability mass 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
- Countable intersection of dense open sets is dense
- Tietze extension theorem
- Continuity of probability measures
- Independence of random variables
- Geometric distribution
- Monotonicity of the cumulative distribution function
- Consistent estimator
- Poisson limit theorem
- Negative binomial distribution
- Tail sum formula for expectation
- 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
- Symmetric difference
- Proper metric space
- Product topology
- Countable union of countable sets is countable
- Probability-generating function
- Memorylessness
- g to the order of the group is the identity
- Conditional independence
- Path-connected spaces are connected
- Central moment
- Operator norm
- Law of large numbers
- Probability space
- Doob's upcrossing inequality
- Relative CW complex
- Riemann domain over C^n
- Univalent function
- Translation surface
- Bernstein–von Mises theorem
- Busemann G-space
- Cramér–Rao bound
- Stone–Weierstrass theorem
- Pettis measurability theorem
- Efficiency of an estimator
- Log-rank test
- Arakelyan's theorem
- Functional data analysis
- Poisson regression
- Sequential analysis
- Locally compact group
- Ergodic decomposition
- Nonexistence of infinite-dimensional Lebesgue measure
- Köthe–Toeplitz dual
- Koksma–Hlawka inequality
- Hodges' superefficient estimator
- Martingale representation theorem
- Classification of surfaces
- Namioka--Phelps
- Mathematical statistics
- F-test
- Laplace transform of the Dirac delta
- Stationary process
- Orthogonal polynomials on the unit circle
- Lagrange's theorem
- Nilradical is the intersection of the prime ideals
- L2 convergence of Fourier series
- Cramér–Wold theorem
- Ultrafilter lemma
- Brouwer's characterization of the Cantor space
- Boolean model
- Convergence of the Neumann series for Volterra kernels
- Exponential of a bounded operator
- Kolmogorov–Arnold–Moser theorem
- Schwarz–Christoffel mapping
- Stochastic partial differential equation
- 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
- Empirical Bayes method
- Karhunen–Loève theorem
- Pettis integral
- Strongly measurable function
- Brent's method
- Mergelyan's theorem
- Itô isometry
- CW complex
- High indices theorem
- E-infinity operad
- Hille–Yosida theorem
- Bayes factor
- Diaconescu's theorem
- Pickands–Balkema–De Haan theorem
- Global attractor
- Dynkin system
- Runge's theorem
- Basu's theorem
- Bump function
- Vanishing mean oscillation
- Jacobson topology
- Spectral decomposition theorem
- Banach bundle
- Variogram
- Robbins–Monro algorithm
- Contiguity of measures
- Order statistic
- Dirichlet process
- Absolute neighborhood retract
- Lindley's paradox
- Tychonoff space
- Universally measurable set
- M-estimator
- Kunen's inconsistency theorem
- Ornstein–Uhlenbeck process
- Kernel regression
- Bayes estimator
- 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
- Neumann series
- Kolmogorov automorphism
- Standard map
- Score
- Hall's theorem on solvable groups
- F-distribution
- Δ-complex
- Lebesgue covering dimension
- Edgeworth expansion
- Crofton formula
- Bourbaki–Witt theorem
- Continuous functional calculus
- Hausdorff paradox
- Monotone likelihood ratio
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Metropolis–Hastings algorithm
- Minimum total potential energy principle
- Continuous-time Markov chain
- Banach limit
- Kriging
- Identifiability
- Baire function
- Branched covering
- Minimal sufficient statistic
- Semiring of sets
- Order dimension
- Pi-system
- Whittle likelihood
- Logistic regression
- Carathéodory's theorem on boundary extension
- Convergence to equilibrium of ergodic Markov chains
- Cramér–von Mises criterion
- Azuma's inequality
- Fréchet manifold
- Gaussian free field
- Parry measure
- Kernel density estimation
- Proof that the order of an element divides the order of the group
- Algebra of sets
- Arnold's cat map
- Erdős–Kac theorem
- Detailed balance
- Coupling from the past
- Robust statistics
- Hidden Markov model
- Finite-dimensional distributions
- Continuous mapping theorem
- Bias–variance decomposition
- Lévy process
- Wald's equation
- Random ordinary differential equation
- Topological ring
- Birkhoff's subdirect representation theorem
- Suslin's problem
- Hadamard gap theorem
- Monte Carlo integration
- Time series
- Optional stopping theorem
- Measurable space
- Singular simplicial set
- Breakdown point
- Delta method
- Regular surface
- Minimum description length
- Hall subgroup
- Noncentral chi-squared distribution
- Characterizations of semisimple modules
- Hájek–Le Cam convolution theorem
- Harmonic measure
- Behrens–Fisher problem
- FK-space
- Anosov flow
- Bochner integral
- Submartingale
- Hodges–Lehmann estimator
- Kruskal–Wallis test
- Midpoint convex function
- Stochastic difference equation
- Sampling distribution
- 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
- Bayes risk
- Statistical inference
- Fredholm map
- Minimum-variance unbiased estimator
- Frattini subgroup
- Sign test
- Bounded sequences ℓ∞
- 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
- Space of convergent sequences
- Sinai billiard
- Liouville–Neumann series
- Geometric Brownian motion
- Empirical likelihood
- Topological field
- Dvoretzky–Kiefer–Wolfowitz inequality
- Bayesian information criterion
- Dvoretzky–Rogers theorem
- Berry–Esseen theorem
- Continuity of measure from below and above
- Hunt process
- Cochran's theorem
- Stochastic approximation
- Lusternik–Schnirelmann category
- Kronecker's approximation theorem
- Jackknife
- Quantile regression
- Markov chain mixing time
- Birkhoff's ergodic theorem
- Akaike information criterion
- Schauder fixed-point theorem for integral equations
- Milnor's isospectral flat tori
- Poincaré inequality
- Hewitt–Savage zero–one law
- Jensen's inequality for integrals
- Statistical model
- Berkovich space dynamics
- Reflection principle
- Credible interval
- Conformal modulus
- Contraction mapping principle for integral equations
- Density of trigonometric polynomials in continuous functions
- Completion of a valued field
- Orbifold
- Ill-posedness of first-kind equations
- Independence of sample mean and sample variance for normal samples
- Self-similar process
- Gibbs sampling
- Estimating equations
- Spectral radius formula
- Supermartingale
- Anderson–Darling test
- Spearman's rank correlation coefficient
- Family-wise error rate
- Subshift of finite type
- Torus knot
- Markov chain Monte Carlo
- Suslin line
- Nested intervals theorem
- Compactness of integral operators
- Hankel operator
- Dickey–Fuller test
- Orientation of a manifold
- 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
- Semimartingale
- Empirical distribution function
- Bootstrap
- Curtis–Hedlund–Lyndon theorem
- Influence function
- Orlicz–Pettis theorem
- Oseledets theorem
- Itô calculus
- Alexander subbase theorem
- Ancillary statistic
- Commutative Gelfand–Naimark theorem
- Stopped process
- Jeffreys prior
- Transcendence degree
- Fourier basis of L2 of the circle
- Hyperbolic set
- Nerve of a cover
- Moments of the log-normal distribution
- Hilbert manifold
- Jentzsch's theorem on positive kernels
- CW approximation
- Complete statistic
- Horvitz–Thompson estimator
- Quasisymmetric map
- Liouville–Arnold theorem
- U-statistic
- Fractional Brownian motion
- Bayesian hierarchical model
- L-estimator
- Countable additivity
- Power of a test
- Parametric model
- Banach–Stone theorem
- Gödel's consistency of CH and AC
- Banach–Mazur theorem
- Ring of sets
- Randomization
- Sufficient statistic
- Gaussian process
- Normal family
- Missing data mechanisms
- Likelihood-ratio test
- Cardinal characteristics of the continuum
- Karlin–Rubin theorem
- Noncommutative geometry
- Schauder basis
- Axiom A
- Tsirelson space
- Conjugate prior
- Kolmogorov–Smirnov test
- Ladyzhenskaya–Babuška–Brezzi condition
- Singular homology
- Delta derivative on a time scale
- Expectation–maximization algorithm
- Ultrafilter
- Hotelling's T-squared distribution
- Boolean prime ideal theorem
- Burkholder–Davis–Gundy inequality
- Doob–Meyer decomposition theorem
- Resolvent as Laplace transform of a semigroup
- Cross-validation
- Barbier's theorem
- Wald test
- Szpilrajn extension theorem
- Extremal length
- Pivotal quantity
- Banach algebra
- Generalized linear model
- Random graph
- Iterated function system
- Tame Fréchet space
- 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
- Shrinkage estimator
- Graph embedding
- Pesin theory
- Thom–Mather stratified space
- Cantor algebra
- Empirical process
- Convenient vector space
- Binomial proportion confidence interval
- Lévy's upward theorem
- Proof of Lagrange's theorem
- Existence of transcendence bases
- False discovery rate and the Benjamini–Hochberg procedure
- Particle filter
- Cantor space
- Neyman construction
- Axiom of determinacy
- Space of null sequences
- Natural boundary
- Riemann surface
- Unique solvability of Volterra equations of the second kind
- Wilks' theorem
- Vitali set
- Pseudo-Anosov map
- Anosov diffeomorphism
- Analytic capacity
- Mittag-Leffler's theorem
- Algorithmic randomness
- Stability conjecture
- Geometric realization
- Skorokhod's embedding theorem
- Prime end
- Torus as a group
- Blaschke–Lebesgue theorem
- Semi-continuity
- 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
- Frequentist inference
- Likelihood principle
- Bunimovich stadium
- Maximal torus
- Fisher–Neyman factorization theorem
- Girsanov theorem
- Cauchy's functional equation
- Quadratic variation
- Faber polynomials
- Observed information
- Feller process
- Point estimation
- Montel's theorem
- Local martingale
- Completion of a measure
Further reading
- Web https://en.wikipedia.org/wiki/Union_(set_theory), lead section
About this page
| Tag | 07KVB4JSC7 |
|---|---|
| URI | m2://07KVB4JSC7 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 03E20 |
| Defines | Union |
| Uses | Set, Element, Disjunction |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.