Binary relation

Binary relation
Tag	`08EJC6Q25H`
URI	`m2://08EJC6Q25H`
Kind	definition
Writing	W1 stated
MSC	03E20
Defines	Binary relation
Uses	Cartesian product, Subset

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Definition

Let $A$ and $B$ be sets. A binary relation from $A$ to $B$ is a subset $R$ of the Cartesian product $A \times B$ . For $a \in A$ and $b \in B$ , write $a \mathrel{R} b$ when $(a, b) \in R$ . A binary relation on $A$ is a binary relation from $A$ to $A$ .

Remarks

Examples on $\mathbb{Z}$ are $\le$ , $=$ and divisibility $\mid$ . A relation on a finite set can be drawn as a directed graph, with an arrow from $a$ to $b$ when $a \mathrel{R} b$ . A function is a special kind of binary relation.

Explanations

L1 human

A relation is a rule that says which pairs belong together. For the relation "divides" on $\{1, 2, 3, 4\}$ , the pair $(2, 4)$ is in the relation and the pair $(3, 4)$ is not.

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

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

Closure is the set of limits of sequences 003QP7A8HG
Variance 007QED3E0Y
Finite intersection property 00BWV5CKCF
Hasse diagram 00EYM7MDKS
Compact set 00R0X1WGPX
Second-countable space 00RE1R4W7N
Partial order 011C29Q1S1
Chernoff bound 014PEBGDYF
Chapman–Kolmogorov equation 01APZ8GDQH
Bias of an estimator 01BD2TFFP4
Totally disconnected space 01CYW8S04D
Urysohn's lemma 01EY3VR7ZD
Composition of injections is injective 01FPXMTBX1
Order of cardinals 01GTP8M5RG
Equivalence classes form a partition 01HT2CH8BH
Triangle inequality for a metric 01M6CW9VM6
Antisymmetric relation 01MGYYBB9R
Infinite set 01MKW4R283
Isolated point 01PKQYTTMT
Composition of relations 01SWBKDAWA
Bijective iff it has a two-sided inverse 01VG2A0BHW
Cantor set is closed 01VT24J1SD
Identically distributed random variables 01X8YSDTBW
Open set 01ZQQDCERP
Law of the unconscious statistician 01ZWX875AT
Well-founded relation 0230ECX1Q5
Left inverse 02486SD328
Density of the normal distribution 027JF1FBB0
Exponential distribution 028Z8WC5Y9
Covariance matrix 02CBHV4ZZQ
Open balls form a basis of the metric topology 02EEX96DFS
Chebyshev's inequality 02HZ1BK3MW
Isometries are injective 02J4G4J4TJ
Characteristic function of a sum of independent random variables 02JGYG0T7M
Stationary distribution 02KYGSCTCN
Closure is closed 02TZDKG50Q
Bernoulli process 02XPXR1FBK
Total order 0316AYH1SC
Markov chain 033TJBZDG0
Statistic 036756NFD6
Arzelà–Ascoli theorem 0381S8Z5P0
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
Lebesgue number lemma 040CFNJEAP
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism 0430ETCMKD
Product metric 044D4AQV8V
Möbius strip 04AJE0M271
Correlation coefficient 04CTY6XYWN
Variance under translation 04CYN5G6RY
Almost sure convergence 04P7D63W8P
Law of total expectation 04RRB6HM00
Zorn's lemma 04RXTQA5ZA
Probability density function 04YG690N5V
The real numbers are uncountable 0518MM6BE1
Compact iff complete and totally bounded 054Q5PW5GX
Expectation of a product of independent random variables 05AW1G0WE1
Closed ball 05DSV40EB9
Likelihood function 05FSTXM92W
T1 space 05GQJXBP3F
Homeomorphism 05GZ11PWRY
Composition of continuous maps is continuous 05NK26G0DR
Markov's inequality 05PWWHYBX5
Standard score 05RZ4YZG4E
Poisson point process 05TG0SE4HT
Gamma distribution 06395GBNWD
Metric space 063W5JMY85
Inclusion of a subspace is continuous 065J822CKN
De Moivre–Laplace theorem 065X9S8CM3
Closed subsets of compact sets are compact 066HW6D9A7
The integers are countable 066RJJVWH9
Closure in a metric space 068P0P32DW
Diameter of a set 0692CGQF00
Heine–Cantor theorem 06B8BNJCD6
Convergence of a sequence in a metric space 06JYPJ4WHH
Kolmogorov's zero–one law 073Q8Z94KG
Correspondence between equivalence relations and partitions 074482Q7CD
The reals have the cardinality of the continuum 075PM9FH7A
Independent random variables are uncorrelated 077GN9HDR6
Convolution of probability distributions 07CHJ5YX28
Lipschitz maps are uniformly continuous 07DCKDXRBP
Composition of surjections is surjective 07EH4R3RVQ
Function composition 07HF9YCWPP
Continuity at a point of a topological space 07N7W5C29X
Heine–Borel theorem 07PFYV65MB
Normal distribution 07RXBVTVZB
Distance from a point to a set 07TXYYVYZR
Absolutely convergent series in a Banach space converge 082MGNWNA8
Equivalence class 082RMRSY26
Mean of the Poisson distribution 086HEVXR9Z
Isometry 0878DN3WCS
Statistical hypothesis test 087P03NEND
Equinumerous sets 08EJHHMB5R
Complete metric space 08EWT7J0SC
Product of countable sets is countable 08G4XH86JC
Sequence spaces lp 08P6T2WH6R
Independence means the joint law is the product of the marginals 08QCA2F1NX
Topological embedding 08SVRVHJNY
Median 08W2CW8PKW
Domain 093CM69AGS
Riesz's lemma 097C7KCAT5
Intermediate value theorem for connected spaces 098C0KZZAP
Dirac measure 099SXAG014
Open sets characterised by neighbourhoods 09CJ00H3NW
Closed sets contain their limits 09GQ3ZTQJ9
Completeness of the space of bounded continuous functions 09GRCZVR5E
Method of moments 09J1H07VAH
Metrizable space 09KDHB82SG
Finite intersection of open sets is open 09NDE9V7X1
Mean of the normal distribution 09PR1V6RWR
Intersection of closed sets is closed 09YMV8H4X5
Product of compact sets is compact 09ZKE5XQ74
Open sets of a metric space 0A26X79RTB
Metric topology 0A6YRNZB2R
Baire category theorem 0A7KGAYHGV
Schröder–Bernstein theorem 0A9V6RKQG6
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
Marginal distribution 0AXP2Z5N9H
Moment-generating function of the normal distribution 0B1G4E21G7
Open balls are open 0B2305GNAZ
Quantile function 0B3Q3XDM5X
Continuity via closed sets 0B8GPXFSP7
Connected subsets of the real line are intervals 0B9ZQFMQNV
Finite set 0BBTH3YSZA
Banach space 0BDTTM24DP
Weak law of large numbers 0BFDHF2H7N
Random walk 0C08YMV95H
Linear maps on finite-dimensional spaces are continuous 0C13E623WD
Random variable with a density 0C21RZG5V2
Interior 0C38S0KMJK
Uncorrelated random variables that are not independent 0C6J8BQXVR
Kurtosis 0CBQVVNBMP
Martingale 0CDVK9RQBH
Transitive relation 0CYRECG2YQ
Dense set 0D198GPMTJ
Restriction of a function 0D2MXBV21T
A set is closed iff it equals its closure 0D2W7FS52T
Baire space 0D5KR33H7X
Variance of a sum 0D86GJEZ39
Limits are unique in Hausdorff spaces 0D8T984XEE
Connected space 0D939CMME7
Mean of the geometric distribution 0DB0R4N5XX
Open ball 0DC3MPR7C9
Expected value 0DCT0QZVQ1
Right inverse 0DDPZAG5ER
Strong law of large numbers 0DFWHQ839T
Covariance 0DJ8W3HP2D
Random variable 0DKRR22AEQ
Closure 0DRKT1QJR7
The rationals are countable 0DS9KWFDK2
Independence of two random variables 0E9502NT92
Range of a function 0E9GTAV478
Neyman–Pearson lemma 0EE19CFEEF
Convergence in probability 0EFDDXTVD6
P-value 0EHYZFZ4PK
Epsilon-delta continuity in metric spaces 0EKGKY5TGH
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
Locally compact space 0EQMWQ4WMJ
68–95–99.7 rule 0EYQ03XCR3
Cantor's intersection theorem 0EZBG8SKJM
Well-order 0F1D6PK9RZ
Transition matrix 0F1KBVY161
Conditional variance 0F1SN93KXE
Bounded linear maps are continuous 0F3K4C7XYW
Codomain 0F9JHP8AJN
Closed subsets of complete spaces are complete 0FBXJZX3MK
Universal property of the quotient topology 0FF2FJ8CH2
Preimage of a set 0FGCFY1AZX
Epsilon-delta uniform continuity 0FHSDCQACY
Lévy's continuity theorem 0FPWF7TTMB
Convergence of random variables 0FQ968YKA1
Meagre set 0FQPYAY9PJ
Maximum likelihood estimation 0FREPFKNS7
Identity function 0FSNQ0GNCC
Completion of a metric space 0G4J7Z7WPJ
Multivariate normal distribution 0G558GJQMW
Closed set 0G5AV9Q4RR
Lexicographic order 0G77XFBV00
Separable space 0GKJ78948V
Antichain 0GNGPF7KS5
Chain 0GP9ASVQ1X
Cardinality of the power set 0GPFKS96T7
Skewness 0GQ3H7VW3V
Pseudometric space 0H3W614GEC
Upper bound 0H5J02P8S9
Galton–Watson process 0H631CZP5K
Interior is the largest open subset 0HAZEFH137
Markov property 0HB8SXHPFA
Limit point 0HBQC2GJXT
Union of open sets is open 0HEF46RPEK
First-countable space 0HG11FW8MJ
Axiom of choice 0HTGB13FE2
Injective iff it has a left inverse 0HYFEEZAEM
Student's t-test 0HYP606RYF
Equivalent norms 0J3578562S
Associativity of composition 0J6AKYZQT8
Discrete random variable 0J6Z6H5VQJ
Bernoulli distribution 0J97P5KZY0
Converse relation 0JCSQNNP2M
Convergence in distribution 0JDAQHWA9N
The cumulative distribution function determines the distribution 0JEVZAFDY6
Regular space 0JF9KW2TXB
Boundary 0JJ9J1XZ8G
Sample variance 0JN9P4A944
Cardinality 0JTGRV4WT4
Tychonoff's theorem 0JTRSJKCFB
Bayesian inference 0JWRJX2RMP
Path-connected space 0K0GE6BCYW
Topologist's sine curve 0K3T4A659R
Confidence interval 0K4NMMJBG2
Complement of an open set is closed 0K67Q60Z8P
Compactness and sequential compactness agree in metric spaces 0K8SGQSV7G
Continuous image of a compact set is compact 0K9AAJ8G51
Hausdorff maximal principle 0KM4KVR9MV
p-norm on R^n 0KNJWEVSMG
Nowhere dense set 0KSM23SN3H
Variance under scaling 0KVQ9D86BM
Discrete uniform distribution 0KYP951QWH
Universal property of the product topology 0M6W4XNST6
Epsilon characterisation of Cauchy sequences 0MCK8K2D7Z
Minimal element 0MCR8ZAXQD
One-point compactification 0MM45BT6HC
Continuum hypothesis 0MN2MER9WN
Compact Hausdorff spaces are normal 0MQQQ5NA2S
Contraction mapping 0MT2S0ZX4Y
Equicontinuity 0MW2WXTHMC
Hausdorff space 0MXB4CKHFW
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
Well-ordering theorem 0N109NG5ZX
Completeness via Cauchy sequences 0N1ZWFJDN5
Hilbert's hotel 0N3Y8G340X
Symmetric relation 0N9SA25Z3V
Cauchy distribution 0NA6Q82YR7
Variance as the mean of the square minus the square of the mean 0NMF95YK8W
Epsilon-net characterisation of total boundedness 0NMGJJ19HD
Birthday problem 0NMWG2PHJ0
Moment-generating function 0NNRNGKMFM
Cantor pairing function 0NP5N5PYW5
Student's t-distribution 0NQ5798SB1
Connected components are closed 0NRT0E2QT2
Characteristic function of the normal distribution 0NS0TDA4E2
Open cover 0NVPNC12JY
Sum of independent normal random variables 0NXGPJWNFN
Closed map 0NYZ2M45G9
Compactness via open covers 0P6PMJB4MA
Linearity of expectation 0P7PTC8DDE
Bounded set 0P953MG05E
Metric spaces are Hausdorff 0P9DCDES73
Sup norm on bounded continuous functions 0PARCKE8WP
Transitive closure 0PEJH7AC74
Moment 0PH0VHTQS0
Heine–Borel theorem in finite-dimensional normed spaces 0PHT6NBYPB
Projections are continuous 0PM88Z9W4M
Interior is open 0PNWCM559P
Beta distribution 0PP5ZAJPP7
Expectation with respect to the binomial distribution 0PVCTCKD3C
Injective function 0PVG1CRAJN
Cumulant generating function 0Q2QC81VTW
Image of a set 0Q3F20XB0A
Joint probability distribution 0QAR0XDZH4
Equivalence relation induced by a function 0QE5PFJY7V
Stochastic process 0QH5ENKZM1
Cardinal number 0QKPNZKNEA
Sample mean 0QNRBHWV35
Compact space 0QPHW703PV
Connected set 0R37DFJKFW
T0 space 0R3B7DX55J
Type I and type II errors 0R41PVFA39
Compact subsets of a metric space are bounded 0R7BTEWCJ2
Ordinal number 0RDJK7EQYY
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
Every infinite set has a countably infinite subset 0RWQ5VZ6VQ
Image of an intersection 0RX7DDBYCK
Preorder 0RXS8R6K37
Continuous linear map 0S3SMRH0Z5
Path component 0SB9FNZECG
Countably infinite set 0SEZ3JGZP7
Equivalence relation 0SGTMBEWRF
Closure is the smallest closed superset 0SHC6Z304Z
Bijective function 0SNQEF7Z53
Hypergeometric distribution 0SPBPW3K61
Open map 0SQ2PC596Y
Moments are derivatives of the moment-generating function at zero 0STK8KEY9G
Almost sure convergence implies convergence in probability 0SVN6X695K
Continuous map of topological spaces 0SY9ECQ19Q
Bertrand paradox 0T0JWJ3M47
Least element 0T2NTJCV83
Subspace metric 0T2WYDTRP2
The characteristic function determines the distribution 0T8BYAFEXX
Lipschitz map 0TMZ21WKCN
Law of a random variable 0TST4MHA3A
Convergence in mean 0TTA57SEJG
Estimator 0TTNRP5PFQ
Equivalent metrics 0TWQ5PRKWS
Surjective iff it has a right inverse 0TYQNHVB4C
Maximal element 0TZEEFTNCC
Variance of a sum of independent random variables 0V0778DCW9
Aleph-null 0V2H5PGJ9B
Continuous real function on a compact set attains its bounds 0V4DET59NW
Mean squared error 0V4VSCBXP3
Quotient topology 0V8046R0P9
Poisson distribution 0VEFSSKX18
Continuity is equivalent to sequential continuity in metric spaces 0VGPKZ8MRK
Banach fixed point theorem 0VK6VEH345
Isometric isomorphism 0VRD8GC5BM
Binomial distribution 0VRWZDWW6B
Indicator function 0VSB0XF4G7
Standard deviation 0VTN5RKNC5
Compact closed unit ball implies finite dimension 0VZVV8VME5
Complete subspaces are closed 0W41XG4W60
Inverse function 0W766JAPSN
Cardinality of the continuum 0WC28W9DNM
Totally bounded set 0WE5BBHGM6
Probability mass function 0WF4EJP91W
Monotone function 0WFF6AZPBP
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
Cantor's theorem 0WTEMGYV2C
Hölder continuity 0WZTM0EFR6
Countable intersection of dense open sets is dense 0X2T63GFQQ
Tietze extension theorem 0X52TNMTCJ
Uncountable set 0X55M9XET4
Independence of random variables 0XCE070V8B
Cantor's diagonal argument 0XCH8QVDCQ
Geometric distribution 0XCVV5Q500
Monotonicity of the cumulative distribution function 0XD22PBMWA
Reflexive relation 0XEGNW851S
Function 0XHDTE5YMZ
Consistent estimator 0XJVF81QF6
Surjective function 0XKH881BZ7
Poisson limit theorem 0XKTZRWTXX
Negative binomial distribution 0XN7KEVSSS
Graph of a function 0Y0KFHF47G
Tail sum formula for expectation 0Y6QNF1J49
Order isomorphism 0Y74KN99A5
Conditional expectation 0YAMJRK04E
Absorbing Markov chain 0YB93GWTKK
Comparability of cardinals 0YBHSJMN3J
Continuity via open sets 0YF82GBTND
Sum of independent Poisson random variables 0YH1M56SBN
Closed balls are closed 0YS40B38JD
Conditional probability distribution 0YVDC5PYDW
Choice function 0YXEQWRGP0
Proper metric space 0Z4V98VN5E
Greatest element 0Z7HDDXX3X
Countable union of countable sets is countable 0Z8FM54P86
Probability-generating function 0Z8PSSAVBA
Memorylessness 0ZAZZ34WQS
Countable set 0ZG61TFBES
Transfinite induction 0ZHQMQ2C1N
Path-connected spaces are connected 0ZMT1GP4KM
Central moment 0ZT24EC90K
Operator norm 0ZTEH055YF
Law of large numbers 0ZVEAYVRC3
Quotient set 0ZVHXDSZC7

Sources

Web https://en.wikipedia.org/wiki/Binary_relation, lead section