Complement

Complement
Tag	`0B1FSS1VZW`
URI	`m2://0B1FSS1VZW`
Kind	definition
Writing	W1 stated
MSC	03E20
Defines	Complement
Uses	Universal set, Subset, Element

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Definition

Let $U$ be a universal set and let $A$ be a subset of $U$ . The complement of $A$ in $U$ is the set $A^{c} = \{x \in U \mid x \notin A\}.$

Remarks

Other notations are $\overline{A}$ , $A'$ and $U \setminus A$ . Example: in $U = \mathbb{Z}$ , the complement of the set of even integers is the set of odd integers. The complement of $A^c$ is $A$ , and $A \cup A^c = U$ , $A \cap A^c = \varnothing$ . The complement of $A$ in $U$ is the set difference $U \setminus A$ .

Explanations

L1 human

The complement is "everything else". It depends on the universe: the complement of the even numbers is the odd numbers in $\mathbb{Z}$ , but not in $\mathbb{R}$ .

What this node 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 node is. They do not measure its truth.

What depends on this

235 nodes use this node, directly or through other nodes.

Closure is the set of limits of sequences 003QP7A8HG
Variance 007QED3E0Y
Finite intersection property 00BWV5CKCF
Prior probability 00NBBSXRQS
Event 00SQGJNRTB
Chernoff bound 014PEBGDYF
Chapman–Kolmogorov equation 01APZ8GDQH
Bias of an estimator 01BD2TFFP4
Totally disconnected space 01CYW8S04D
Urysohn's lemma 01EY3VR7ZD
Cantor set is closed 01VT24J1SD
Identically distributed random variables 01X8YSDTBW
Law of the unconscious statistician 01ZWX875AT
Density of the normal distribution 027JF1FBB0
Exponential distribution 028Z8WC5Y9
Covariance matrix 02CBHV4ZZQ
Chebyshev's inequality 02HZ1BK3MW
Characteristic function of a sum of independent random variables 02JGYG0T7M
Stationary distribution 02KYGSCTCN
Closure is closed 02TZDKG50Q
Bernoulli process 02XPXR1FBK
Markov chain 033TJBZDG0
Statistic 036756NFD6
Law of total variance 03BY5XTNA4
Variance of the normal distribution 03EESZDB9A
Equivalence of norms in finite dimension 03MC70ZRNR
Torus 03NM7DZXXC
Cauchy–Schwarz inequality for expectations 03PGHKAJG3
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism 0430ETCMKD
Monotonicity of probability 0464Z0JP2E
Correlation coefficient 04CTY6XYWN
Variance under translation 04CYN5G6RY
Almost sure convergence 04P7D63W8P
Law of total expectation 04RRB6HM00
Sigma-algebra 04WSW8T3SY
Probability density function 04YG690N5V
Expectation of a product of independent random variables 05AW1G0WE1
Likelihood function 05FSTXM92W
T1 space 05GQJXBP3F
Markov's inequality 05PWWHYBX5
Standard score 05RZ4YZG4E
Poisson point process 05TG0SE4HT
Gamma distribution 06395GBNWD
De Moivre–Laplace theorem 065X9S8CM3
Closed subsets of compact sets are compact 066HW6D9A7
Closure in a metric space 068P0P32DW
Buffon's needle problem 06X5PF6YFP
Kolmogorov's zero–one law 073Q8Z94KG
Independent random variables are uncorrelated 077GN9HDR6
Convolution of probability distributions 07CHJ5YX28
Heine–Borel theorem 07PFYV65MB
Normal distribution 07RXBVTVZB
Mean of the Poisson distribution 086HEVXR9Z
Statistical hypothesis test 087P03NEND
Sequence spaces lp 08P6T2WH6R
Independence means the joint law is the product of the marginals 08QCA2F1NX
Median 08W2CW8PKW
Geometric probability 0924BCD257
Riesz's lemma 097C7KCAT5
Intermediate value theorem for connected spaces 098C0KZZAP
Dirac measure 099SXAG014
Closed sets contain their limits 09GQ3ZTQJ9
Method of moments 09J1H07VAH
Mean of the normal distribution 09PR1V6RWR
Independent sigma-algebras 09T8ZHST1V
Intersection of closed sets is closed 09YMV8H4X5
Baire category theorem 0A7KGAYHGV
Finite-dimensional normed spaces are complete 0ABP2F2ZP7
Metrizable spaces are normal 0AD0BZ49DW
Normal space 0AJ646N339
Independent and identically distributed random variables 0AJN7WHBJZ
Classical definition of probability 0AKNPYQ04R
Pairwise independent events that are not mutually independent 0AR4B3K5QD
Marginal distribution 0AXP2Z5N9H
Moment-generating function of the normal distribution 0B1G4E21G7
Quantile function 0B3Q3XDM5X
Continuity via closed sets 0B8GPXFSP7
Connected subsets of the real line are intervals 0B9ZQFMQNV
Weak law of large numbers 0BFDHF2H7N
Multiplication rule 0BS3RMX8HM
Borel sigma-algebra 0BVHZXWKCD
Addition rule for two events 0BZQ6ZX584
Random walk 0C08YMV95H
Linear maps on finite-dimensional spaces are continuous 0C13E623WD
Random variable with a density 0C21RZG5V2
Uncorrelated random variables that are not independent 0C6J8BQXVR
Kurtosis 0CBQVVNBMP
Martingale 0CDVK9RQBH
Dense set 0D198GPMTJ
A set is closed iff it equals its closure 0D2W7FS52T
Baire space 0D5KR33H7X
Variance of a sum 0D86GJEZ39
Connected space 0D939CMME7
Probability axioms 0DAX8PMAJ4
Mean of the geometric distribution 0DB0R4N5XX
Expected value 0DCT0QZVQ1
Strong law of large numbers 0DFWHQ839T
Covariance 0DJ8W3HP2D
Random variable 0DKRR22AEQ
Closure 0DRKT1QJR7
Base rate fallacy 0E44JMG6NV
Law of total probability 0E5G5FMT2R
Independence of two random variables 0E9502NT92
Neyman–Pearson lemma 0EE19CFEEF
Convergence in probability 0EFDDXTVD6
Boole's inequality 0EGSEE9J9R
P-value 0EHYZFZ4PK
Chi-squared test 0EN0V3NY20
Continuous image of a connected set is connected 0EPD87YVG8
Convergence in probability implies convergence in distribution 0EPTDZPJ83
Central limit theorem 0EPZC4Z1FC
68–95–99.7 rule 0EYQ03XCR3
Cantor's intersection theorem 0EZBG8SKJM
Transition matrix 0F1KBVY161
Conditional variance 0F1SN93KXE
Closed subsets of complete spaces are complete 0FBXJZX3MK
Conditional probability 0FJB06AD2P
Lévy's continuity theorem 0FPWF7TTMB
Convergence of random variables 0FQ968YKA1
Meagre set 0FQPYAY9PJ
Maximum likelihood estimation 0FREPFKNS7
Completion of a metric space 0G4J7Z7WPJ
Multivariate normal distribution 0G558GJQMW
Closed set 0G5AV9Q4RR
Mutual independence of events 0GCVEV6BJA
Separable space 0GKJ78948V
Skewness 0GQ3H7VW3V
Galton–Watson process 0H631CZP5K
Markov property 0HB8SXHPFA
Student's t-test 0HYP606RYF
Discrete random variable 0J6Z6H5VQJ
Bernoulli distribution 0J97P5KZY0
Convergence in distribution 0JDAQHWA9N
The cumulative distribution function determines the distribution 0JEVZAFDY6
Regular space 0JF9KW2TXB
Boundary 0JJ9J1XZ8G
Sample variance 0JN9P4A944
Bayesian inference 0JWRJX2RMP
Topologist's sine curve 0K3T4A659R
Confidence interval 0K4NMMJBG2
Complement of an open set is closed 0K67Q60Z8P
Venn diagram 0KCT6DPXJK
Posterior probability 0KJ7ZS8AJA
p-norm on R^n 0KNJWEVSMG
Nowhere dense set 0KSM23SN3H
Variance under scaling 0KVQ9D86BM
Discrete uniform distribution 0KYP951QWH
One-point compactification 0MM45BT6HC
Compact Hausdorff spaces are normal 0MQQQ5NA2S
Coupon collector's problem 0MXCP7MXWJ
Cumulative distribution function 0MXMS3AE13
Open connected subsets of R^n are path-connected 0MXN7AFBPE
Characteristic function 0MYDNRT7PP
Compact subsets of a Hausdorff space are closed 0MYFJVMN7M
Pairwise independence 0N57D1XY6T
Cauchy distribution 0NA6Q82YR7
Bayes' theorem 0NM6E9VW04
Variance as the mean of the square minus the square of the mean 0NMF95YK8W
Birthday problem 0NMWG2PHJ0
Moment-generating function 0NNRNGKMFM
Student's t-distribution 0NQ5798SB1
Connected components are closed 0NRT0E2QT2
Characteristic function of the normal distribution 0NS0TDA4E2
Second Borel–Cantelli lemma 0NWXRAMEE7
Sum of independent normal random variables 0NXGPJWNFN
Closed map 0NYZ2M45G9
Independence of two events 0P0MZ4X682
Linearity of expectation 0P7PTC8DDE
Monty Hall problem 0PBWWQD9KY
Moment 0PH0VHTQS0
Heine–Borel theorem in finite-dimensional normed spaces 0PHT6NBYPB
Beta distribution 0PP5ZAJPP7
Probability of the complement 0PQEBTYDZM
Expectation with respect to the binomial distribution 0PVCTCKD3C
Cumulant generating function 0Q2QC81VTW
Joint probability distribution 0QAR0XDZH4
Probability distribution 0QB913WM5E
Stochastic process 0QH5ENKZM1
Sample mean 0QNRBHWV35
Connected set 0R37DFJKFW
Type I and type II errors 0R41PVFA39
Almost surely 0R46A1XY60
Generating function of a sum of independent random variables 0REMT01N36
Chi-squared distribution 0RFMMBHFTV
Limit of the cumulative distribution function at infinity 0RJ5TZH6G0
Cantor set is homeomorphic to the space of binary sequences 0RQ8DHHR8V
Closure is the smallest closed superset 0SHC6Z304Z
Hypergeometric distribution 0SPBPW3K61
Moments are derivatives of the moment-generating function at zero 0STK8KEY9G
Almost sure convergence implies convergence in probability 0SVN6X695K
Bertrand paradox 0T0JWJ3M47
First Borel–Cantelli lemma 0T6WS7TT77
The characteristic function determines the distribution 0T8BYAFEXX
Law of a random variable 0TST4MHA3A
Convergence in mean 0TTA57SEJG
Estimator 0TTNRP5PFQ
Variance of a sum of independent random variables 0V0778DCW9
Continuous real function on a compact set attains its bounds 0V4DET59NW
Mean squared error 0V4VSCBXP3
Poisson distribution 0VEFSSKX18
Probability measure 0VGMEBVW1M
De Morgan's laws for sets 0VK0W4D5Y5
Binomial distribution 0VRWZDWW6B
Standard deviation 0VTN5RKNC5
Compact closed unit ball implies finite dimension 0VZVV8VME5
Complete subspaces are closed 0W41XG4W60
Law of total probability for an event and its complement 0W62TKMW7V
Probability mass function 0WF4EJP91W
Gambler's ruin 0WFMKK7DFZ
Continuous uniform distribution 0WGDPYR30Q
Circle as a quotient of the real line 0WH1GVBJNH
Hoeffding's inequality 0WPDNXGSSN
Moment-generating function of a sum of independent random variables 0WRZ170YEH
Countable intersection of dense open sets is dense 0X2T63GFQQ
Tietze extension theorem 0X52TNMTCJ
Continuity of probability measures 0X8HBKCSYB
Independence of random variables 0XCE070V8B
Geometric distribution 0XCVV5Q500
Monotonicity of the cumulative distribution function 0XD22PBMWA
Consistent estimator 0XJVF81QF6
Poisson limit theorem 0XKTZRWTXX
Negative binomial distribution 0XN7KEVSSS
Tail sum formula for expectation 0Y6QNF1J49
Conditional expectation 0YAMJRK04E
Absorbing Markov chain 0YB93GWTKK
Sum of independent Poisson random variables 0YH1M56SBN
Closed balls are closed 0YS40B38JD
Conditional probability distribution 0YVDC5PYDW
Probability-generating function 0Z8PSSAVBA
Memorylessness 0ZAZZ34WQS
Conditional independence 0ZJ87ZZ8PK
Path-connected spaces are connected 0ZMT1GP4KM
Central moment 0ZT24EC90K
Law of large numbers 0ZVEAYVRC3
Probability space 0ZYBZ34RT2

Sources

Web https://en.wikipedia.org/wiki/Complement_(set_theory), lead section

What links here

Used by

Sigma-algebra 04WSW8T3SY
Closed set 0G5AV9Q4RR
Venn diagram 0KCT6DPXJK
Probability of the complement 0PQEBTYDZM
De Morgan's laws for sets 0VK0W4D5Y5