Preimage of a set

Preimage of a set
Tag	`0FGCFY1AZX`
URI	`m2://0FGCFY1AZX`
Kind	definition
Writing	W0 stub
MSC	03E20
Defines	Preimage of a set
Uses	Function, Subset

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Definition

Let $f \colon A \to B$ be a function and let $T$ be a subset of $B$ . The preimage of $T$ under $f$ is the set $f^{-1}(T) = \{a \in A \mid f(a) \in T\}.$

Remarks

Another name is the inverse image. The notation $f^{-1}(T)$ makes sense for every $f$ , also when $f$ has no inverse function. Preimages respect all set operations: $f^{-1}(T_1 \cap T_2) = f^{-1}(T_1) \cap f^{-1}(T_2)$ , and the same holds for unions and complements. Example: for $f(x) = x^2$ on $\mathbb{R}$ , $f^{-1}([1, 4]) = [-2, -1] \cup [1, 2]$ .

Explanations

L1 human

The preimage answers the question: which inputs land in $T$ ? For the function that sends a student to their grade, the preimage of $\{A\}$ is the set of students with grade $A$ .

What this node needs for W1

an inventory row: the row of the inventory that the node accounts for (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

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

Variance 007QED3E0Y
Chernoff bound 014PEBGDYF
Chapman–Kolmogorov equation 01APZ8GDQH
Bias of an estimator 01BD2TFFP4
Urysohn's lemma 01EY3VR7ZD
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
Bernoulli process 02XPXR1FBK
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
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism 0430ETCMKD
Möbius strip 04AJE0M271
Correlation coefficient 04CTY6XYWN
Variance under translation 04CYN5G6RY
Almost sure convergence 04P7D63W8P
Law of total expectation 04RRB6HM00
Probability density function 04YG690N5V
Expectation of a product of independent random variables 05AW1G0WE1
Likelihood function 05FSTXM92W
Homeomorphism 05GZ11PWRY
Composition of continuous maps is continuous 05NK26G0DR
Markov's inequality 05PWWHYBX5
Standard score 05RZ4YZG4E
Poisson point process 05TG0SE4HT
Gamma distribution 06395GBNWD
Inclusion of a subspace is continuous 065J822CKN
De Moivre–Laplace theorem 065X9S8CM3
Heine–Cantor theorem 06B8BNJCD6
Kolmogorov's zero–one law 073Q8Z94KG
Independent random variables are uncorrelated 077GN9HDR6
Convolution of probability distributions 07CHJ5YX28
Continuity at a point of a topological space 07N7W5C29X
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
Topological embedding 08SVRVHJNY
Median 08W2CW8PKW
Intermediate value theorem for connected spaces 098C0KZZAP
Completeness of the space of bounded continuous functions 09GRCZVR5E
Method of moments 09J1H07VAH
Mean of the normal distribution 09PR1V6RWR
Finite-dimensional normed spaces are complete 0ABP2F2ZP7
Independent and identically distributed random variables 0AJN7WHBJZ
Marginal distribution 0AXP2Z5N9H
Moment-generating function of the normal distribution 0B1G4E21G7
Quantile function 0B3Q3XDM5X
Continuity via closed sets 0B8GPXFSP7
Weak law of large numbers 0BFDHF2H7N
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
Variance of a sum 0D86GJEZ39
Mean of the geometric distribution 0DB0R4N5XX
Expected value 0DCT0QZVQ1
Strong law of large numbers 0DFWHQ839T
Covariance 0DJ8W3HP2D
Random variable 0DKRR22AEQ
Independence of two random variables 0E9502NT92
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
68–95–99.7 rule 0EYQ03XCR3
Transition matrix 0F1KBVY161
Conditional variance 0F1SN93KXE
Bounded linear maps are continuous 0F3K4C7XYW
Universal property of the quotient topology 0FF2FJ8CH2
Lévy's continuity theorem 0FPWF7TTMB
Convergence of random variables 0FQ968YKA1
Maximum likelihood estimation 0FREPFKNS7
Multivariate normal distribution 0G558GJQMW
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
Sample variance 0JN9P4A944
Bayesian inference 0JWRJX2RMP
Path-connected space 0K0GE6BCYW
Topologist's sine curve 0K3T4A659R
Confidence interval 0K4NMMJBG2
Continuous image of a compact set is compact 0K9AAJ8G51
p-norm on R^n 0KNJWEVSMG
Variance under scaling 0KVQ9D86BM
Discrete uniform distribution 0KYP951QWH
Universal property of the product topology 0M6W4XNST6
Equicontinuity 0MW2WXTHMC
Coupon collector's problem 0MXCP7MXWJ
Cumulative distribution function 0MXMS3AE13
Open connected subsets of R^n are path-connected 0MXN7AFBPE
Characteristic function 0MYDNRT7PP
Cauchy distribution 0NA6Q82YR7
Variance as the mean of the square minus the square of the mean 0NMF95YK8W
Moment-generating function 0NNRNGKMFM
Student's t-distribution 0NQ5798SB1
Characteristic function of the normal distribution 0NS0TDA4E2
Sum of independent normal random variables 0NXGPJWNFN
Linearity of expectation 0P7PTC8DDE
Sup norm on bounded continuous functions 0PARCKE8WP
Moment 0PH0VHTQS0
Heine–Borel theorem in finite-dimensional normed spaces 0PHT6NBYPB
Projections are continuous 0PM88Z9W4M
Beta distribution 0PP5ZAJPP7
Expectation with respect to the binomial distribution 0PVCTCKD3C
Cumulant generating function 0Q2QC81VTW
Joint probability distribution 0QAR0XDZH4
Stochastic process 0QH5ENKZM1
Sample mean 0QNRBHWV35
Type I and type II errors 0R41PVFA39
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
Continuous linear map 0S3SMRH0Z5
Path component 0SB9FNZECG
Hypergeometric distribution 0SPBPW3K61
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
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
Quotient topology 0V8046R0P9
Poisson distribution 0VEFSSKX18
Continuity is equivalent to sequential continuity in metric spaces 0VGPKZ8MRK
Binomial distribution 0VRWZDWW6B
Standard deviation 0VTN5RKNC5
Compact closed unit ball implies finite dimension 0VZVV8VME5
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
Tietze extension theorem 0X52TNMTCJ
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
Continuity via open sets 0YF82GBTND
Sum of independent Poisson random variables 0YH1M56SBN
Conditional probability distribution 0YVDC5PYDW
Probability-generating function 0Z8PSSAVBA
Memorylessness 0ZAZZ34WQS
Path-connected spaces are connected 0ZMT1GP4KM
Central moment 0ZT24EC90K
Operator norm 0ZTEH055YF
Law of large numbers 0ZVEAYVRC3

Sources

Web https://en.wikipedia.org/wiki/Image_(mathematics), Inverse image

What links here

Used by

Inclusion of a subspace is continuous 065J822CKN
Continuity at a point of a topological space 07N7W5C29X
Continuity via closed sets 0B8GPXFSP7
Random variable 0DKRR22AEQ
Continuous image of a connected set is connected 0EPD87YVG8
Continuous map of topological spaces 0SY9ECQ19Q
Quotient topology 0V8046R0P9