Mathematical logic and foundations Definition
Countable set
Builds on Injective function
Definition
A set is countable when there is an injective function .
Remarks
Equivalently, is finite or countably infinite. A nonempty set is countable exactly when there is a surjection from to it. Some authors use "countable" only for countably infinite sets, and "at most countable" for this notion. Subsets of countable sets are countable.
Reading path
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- Element definition
- Proposition concept
- Set definition
- Implication definition
- Predicate definition
- Ordered pair definition
- Existential quantifier definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- Uniqueness quantifier definition
- Binary relation definition
- Function definition
- Injective function definition
- Countable set definition
Where this leads
232 pages build on this one.
- Variance
- Second-countable space
- Chernoff bound
- Chapman–Kolmogorov equation
- Bias of an estimator
- Law of the unconscious statistician
- Covariance matrix
- Chebyshev's inequality
- Characteristic function of a sum of independent random variables
- Stationary distribution
- Bernoulli process
- Markov chain
- Law of total variance
- Variance of the normal distribution
- Cauchy–Schwarz inequality for expectations
- Correlation coefficient
- Variance under translation
- Law of total expectation
- The real numbers are uncountable
- Expectation of a product of independent random variables
- Likelihood function
- Markov's inequality
- Poisson point process
- De Moivre–Laplace theorem
- Independent random variables are uncorrelated
- Convolution of probability distributions
- Mean of the Poisson distribution
- Product of countable sets is countable
- Method of moments
- Mean of the normal distribution
- Baire category theorem
- Moment-generating function of the normal distribution
- Weak law of large numbers
- Uncorrelated random variables that are not independent
- Kurtosis
- Martingale
- Baire space
- Variance of a sum
- Mean of the geometric distribution
- Expected value
- Strong law of large numbers
- Covariance
- The rationals are countable
- Neyman–Pearson lemma
- Chi-squared test
- Central limit theorem
- 68–95–99.7 rule
- Transition matrix
- Conditional variance
- Lévy's continuity theorem
- Convergence of random variables
- Meagre set
- Maximum likelihood estimation
- Separable space
- Skewness
- Galton–Watson process
- Markov property
- First-countable space
- Student's t-test
- Discrete random variable
- Bernoulli distribution
- Sample variance
- Bayesian inference
- Confidence interval
- Variance under scaling
- Discrete uniform distribution
- Coupon collector's problem
- Characteristic function
- Variance as the mean of the square minus the square of the mean
- Moment-generating function
- Student's t-distribution
- Characteristic function of the normal distribution
- Sum of independent normal random variables
- Linearity of expectation
- Moment
- Expectation with respect to the binomial distribution
- Cumulant generating function
- Sample mean
- Generating function of a sum of independent random variables
- Chi-squared distribution
- Hypergeometric distribution
- Moments are derivatives of the moment-generating function at zero
- The characteristic function determines the distribution
- Convergence in mean
- Variance of a sum of independent random variables
- Mean squared error
- Poisson distribution
- Binomial distribution
- Standard deviation
- Probability mass function
- Hoeffding's inequality
- Moment-generating function of a sum of independent random variables
- Countable intersection of dense open sets is dense
- Uncountable set
- Cantor's diagonal argument
- Geometric distribution
- Poisson limit theorem
- Negative binomial distribution
- Tail sum formula for expectation
- Conditional expectation
- Absorbing Markov chain
- Sum of independent Poisson random variables
- Conditional probability distribution
- Countable union of countable sets is countable
- Probability-generating function
- Memorylessness
- Central moment
- Doob's upcrossing inequality
- Cramér–Rao bound
- Pettis measurability theorem
- Efficiency of an estimator
- Poisson regression
- Ergodic decomposition
- Koksma–Hlawka inequality
- Hodges' superefficient estimator
- Martingale representation theorem
- Cramér–Wold theorem
- Boolean model
- Wald–Wolfowitz optimality of the sequential probability ratio test
- Doob's martingale inequality
- Consistency of the maximum likelihood estimator
- Itô isometry
- Bayes factor
- Basu's theorem
- Lindley's paradox
- M-estimator
- Score test
- Kingman's subadditive ergodic theorem
- McDiarmid's inequality
- Lévy's characterization of Brownian motion
- Asymptotic normality of the maximum likelihood estimator
- Score
- F-distribution
- Edgeworth expansion
- Monotone likelihood ratio
- Δ-system lemma
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Metropolis–Hastings algorithm
- Continuous-time Markov chain
- Cantor's isomorphism theorem
- Minimal sufficient statistic
- Whittle likelihood
- Logistic regression
- Convergence to equilibrium of ergodic Markov chains
- Azuma's inequality
- Gaussian free field
- Erdős–Kac theorem
- Detailed balance
- Coupling from the past
- Hidden Markov model
- Continuous mapping theorem
- Bias–variance decomposition
- Wald's equation
- Suslin's problem
- Monte Carlo integration
- Optional stopping theorem
- Breakdown point
- Delta method
- Noncentral chi-squared distribution
- Hájek–Le Cam convolution theorem
- Behrens–Fisher problem
- Submartingale
- Kruskal–Wallis test
- Information geometry
- Recurrence and transience of states
- Continuous function with divergent Fourier series
- Malliavin calculus
- Wishart distribution
- Birth–death process
- Minimum-variance unbiased estimator
- Sign test
- Slutsky's theorem
- Tanaka's formula
- Empirical likelihood
- Bayesian information criterion
- Berry–Esseen theorem
- Cochran's theorem
- Stochastic approximation
- Jackknife
- Markov chain mixing time
- Birkhoff's ergodic theorem
- Akaike information criterion
- Credible interval
- Orbifold
- Aronszajn tree
- Independence of sample mean and sample variance for normal samples
- Gibbs sampling
- Estimating equations
- Supermartingale
- Spearman's rank correlation coefficient
- Markov chain Monte Carlo
- Suslin line
- Generalized method of moments
- Doob's martingale convergence theorems
- Bernstein inequalities
- Semimartingale
- Bootstrap
- Lawvere's fixed-point theorem
- Influence function
- Oseledets theorem
- Itô calculus
- Complete statistic
- Horvitz–Thompson estimator
- U-statistic
- Sufficient statistic
- Missing data mechanisms
- Likelihood-ratio test
- Cardinal characteristics of the continuum
- Conjugate prior
- Expectation–maximization algorithm
- Hotelling's T-squared distribution
- Burkholder–Davis–Gundy inequality
- Doob–Meyer decomposition theorem
- Wald test
- Generalized linear model
- Box–Jenkins method
- Pesin theory
- Binomial proportion confidence interval
- Lévy's upward theorem
- Neyman construction
- Wilks' theorem
- Skorokhod's embedding theorem
- Fisher's exact test
- Doob decomposition theorem
- Frequentist inference
- Likelihood principle
- Fisher–Neyman factorization theorem
- Girsanov theorem
- Observed information
- Point estimation
- Local martingale
Further reading
- Web https://en.wikipedia.org/wiki/Countable_set, lead section
About this page
| Tag | 0ZG61TFBES |
|---|---|
| URI | m2://0ZG61TFBES |
| Kind | definition |
| Writing | W1 stated |
| MSC | 03E10 |
| Defines | Countable set |
| Uses | Injective function |
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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.