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The preimage of an output value is the set of input values that produce . More generally, evaluating f {\displaystyle f} at each element of a given subset A {\displaystyle A} of its domain X {\displaystyle X} produces a set, called the " image of A {\displaystyle A} under (or through) f {\displaystyle f} ".
For example, the preimage of {,} under the square function is the set {,,,}. By definition of a function, the image of an element x of the domain is always a single element of the codomain. However, the preimage f − 1 ( y ) {\displaystyle f^{-1}(y)} of an element y of the codomain may be empty or contain any number of elements.
Let : be any function. If is any set then its preimage := under is necessarily an -saturated set.In particular, every fiber of a map is an -saturated set.. The empty set = and the domain = are always saturated.
Measurable function: the preimage of each measurable set is measurable. Borel function: the preimage of each Borel set is a Borel set. Baire function called also Baire measurable function: obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions.
The set of all such vectors is the row space of A. In this case, the row space is precisely the set of vectors (x, y, z) ∈ K 3 satisfying the equation z = 2x (using Cartesian coordinates, this set is a plane through the origin in three-dimensional space).
The preimage in G of the center of G/Z is called the second center and these groups begin the upper central series. Generalizing the earlier comments about the socle, a finite p-group with order p n contains normal subgroups of order p i with 0 ≤ i ≤ n, and any normal subgroup of order p i is contained in the ith center Z i.
The family consisting only of the empty set and the set , called the minimal or trivial σ-algebra over . The power set of X , {\displaystyle X,} called the discrete σ-algebra . The collection { ∅ , A , X ∖ A , X } {\displaystyle \{\varnothing ,A,X\setminus A,X\}} is a simple σ-algebra generated by the subset A . {\displaystyle A.}
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.