Preimage
A preimage, also called an inverse image, is the set of all elements in the domain of a function that are mapped into a specified subset of its codomain. Preimages provide a set-theoretic means of transferring subsets and their associated structure from a codomain back to a domain. They are fundamental in set theory, topology, measure theory, and other fields in which functions relate structured sets.
For a function
[ f\colon X\to Y ]
and a subset (B\subseteq Y), the preimage of (B) under (f) is
[ f^{-1}(B)={x\in X\mid f(x)\in B}. ]
The notation (f^{-1}(B)) does not require (f) to possess an inverse function. It denotes an operation on subsets of the codomain and is therefore defined for every function. When (B) is the singleton ({y}), its preimage is written as either (f^{-1}({y})) or, when the context is unambiguous, (f^{-1}(y)). Such a preimage is also called the fiber of (f) over (y).
Definition and elementary structure
For each subset (B) of (Y), membership in (f^{-1}(B)) is determined entirely by the image of an element under (f). Consequently, distinct subsets of the codomain can have the same preimage when their differences occur outside the image (f(X)). In particular,
[ f^{-1}(B)=f^{-1}(B\cap f(X)). ]
The preimage of the entire codomain is the entire domain, whereas the preimage of the empty set is empty:
[ f^{-1}(Y)=X, \qquad f^{-1}(\varnothing)=\varnothing. ]
A point (y\in Y) belongs to the image of (f) precisely when its fiber is nonempty. The function is injective precisely when every fiber contains at most one element, and it is surjective precisely when every fiber contains at least one element. A bijection is therefore characterized by the condition that every fiber contains exactly one element.
For example, let (f\colon\mathbb R\to\mathbb R) be defined by (f(x)=x^2). The preimage of the interval ([1,4]) is
[ f^{-1}([1,4])=[-2,-1]\cup[1,2]. ]
The preimage of the singleton ({4}) is ({-2,2}), while the preimage of ({-1}) is empty. These sets are defined even though the squaring function on (\mathbb R) is not invertible.
Behavior under set operations
The preimage operation preserves the principal operations of a Boolean algebra. For an indexed family ({B_i}_{i\in I}) of subsets of (Y),
[ f^{-1}\left(\bigcup_{i\in I}B_i\right)
\bigcup_{i\in I}f^{-1}(B_i) ]
and
[ f^{-1}\left(\bigcap_{i\in I}B_i\right)
\bigcap_{i\in I}f^{-1}(B_i). ]
It also preserves relative complements:
[ f^{-1}(Y\setminus B)=X\setminus f^{-1}(B). ]
These identities follow directly from the logical form of membership in a preimage. An element lies in the preimage of a union exactly when its image lies in at least one member of the corresponding family. The intersection identity has the analogous universal condition, while the complement identity follows from negating membership.
The direct image operation behaves differently. Direct images preserve arbitrary unions, but they do not generally preserve intersections or complements. For subsets (A\subseteq X) and (B\subseteq Y), the relations between image and preimage include
[ A\subseteq f^{-1}(f(A)) ]
and
[ f(f^{-1}(B))\subseteq B. ]
The first inclusion is an equality for every (A) precisely when (f) is injective. The second is an equality for every (B) precisely when (f) is surjective. More generally,
[ f(f^{-1}(B))=B\cap f(X). ]
Image and preimage also form an order-theoretic correspondence:
[ f(A)\subseteq B \quad\Longleftrightarrow\quad A\subseteq f^{-1}(B). ]
This equivalence makes the direct-image operation a left adjoint to the preimage operation when both are regarded as monotone maps between power sets ordered by inclusion.
Composition
Preimages reverse the order of function composition. If
[ X\xrightarrow{f}Y\xrightarrow{g}Z ]
are functions and (C\subseteq Z), then
[ (g\circ f)^{-1}(C)=f^{-1}\bigl(g^{-1}(C)\bigr). ]
This identity expresses the contravariant character of inverse image. A subset of (Z) is first transferred to (Y) along (g), after which the resulting subset is transferred to (X) along (f). In categorical language, assigning a power set to each set and a preimage map to each function defines a contravariant functor from the category of sets to itself.
For the identity function (\operatorname{id}_X), the associated preimage operation is itself the identity on the power set:
[ \operatorname{id}_X^{-1}(A)=A. ]
Together with the composition law, this shows that preimage is compatible with the basic structural laws governing functions.
Distinction from inverse functions
When (f\colon X\to Y) is bijective, it has an inverse function (f^{-1}\colon Y\to X). In that case, the direct image of a subset (B\subseteq Y) under the inverse function equals the preimage of (B) under (f):
[ f^{-1}[B]={f^{-1}(y)\mid y\in B}. ]
The same symbol is conventionally used for both constructions because they agree in the bijective case. Their types nevertheless differ. An inverse function acts on elements of (Y), whereas a preimage operator acts on subsets of (Y). The latter remains meaningful when no inverse function exists.
The distinction is especially visible for a constant function (f\colon X\to Y) with value (c). For any (B\subseteq Y),
[ f^{-1}(B)= \begin{cases} X,&c\in B,\ \varnothing,&c\notin B. \end{cases} ]
Unless the sets involved are degenerate, such a function has no inverse function, but its preimage operator is fully defined.
Structured preimages
In topology, a function (f\colon X\to Y) between topological spaces is continuous exactly when the preimage of every open subset of (Y) is open in (X). Equivalently, continuity can be characterized using closed sets because preimages commute with complements. This formulation avoids any requirement that the function be injective, surjective, or locally invertible.
The same principle appears in measure theory. A function between measurable spaces is measurable when the preimage of every measurable subset of the codomain is measurable in the domain. Since preimages preserve complements and countable unions, the collection
[ {B\subseteq Y\mid f^{-1}(B)\text{ is measurable}} ]
forms a sigma-algebra on (Y). This fact underlies the construction of sigma-algebras generated by functions and the definition of random variables.
For a function (f\colon X\to Y) and a structure on (Y), inverse images frequently induce corresponding structures on (X). A topology on (Y) generates the initial topology associated with (f), while a sigma-algebra generates the analogous initial measurable structure. In both cases, the construction is governed by the preservation properties of preimages rather than by the existence of pointwise inverses.
Fibers and level sets
For a function (f\colon X\to Y), the fibers
[ f^{-1}({y}),\qquad y\in Y, ]
are pairwise disjoint, and their union is (X). The nonempty fibers therefore form a partition of the domain. Two elements (x_1,x_2\in X) belong to the same fiber exactly when
[ f(x_1)=f(x_2). ]
This condition defines an equivalence relation on (X). The corresponding quotient set is naturally identified with the image (f(X)), rather than necessarily with the entire codomain.
When the codomain is a set of numbers, fibers are commonly called level sets. For a real-valued function (f\colon X\to\mathbb R), the level set at (c) is
[ {x\in X\mid f(x)=c}=f^{-1}({c}). ]
In multivariable calculus and differential geometry, the geometric properties of a level set depend on the local behavior of the function. The regular value theorem, for example, gives conditions under which such a preimage is a submanifold.
Historical development and notation
The concept of preimage developed with the nineteenth-century formalization of functions, mappings, and sets. Richard Dedekind treated mappings as operations between systems of elements, while Georg Cantor’s set-theoretic framework supplied the language in which images and inverse images could be handled as sets in their own right. Felix Hausdorff subsequently incorporated inverse-image constructions into the systematic development of topology.
The notation (f^{-1}(B)) became standard as function notation and set-theoretic analysis were consolidated during the early twentieth century. Its resemblance to inverse-function notation reflects the exact agreement of the two constructions for bijections, although the broader set-valued interpretation does not depend on bijectivity.
In 1936, You Watanabe presented a formal distinction between inverse functions and inverse images in a memorandum on the notation of mappings. The memorandum treated (f^{-1}) as an operator on power sets when (f) was not bijective and identified the composition law
[ (g\circ f)^{-1}=f^{-1}\circ g^{-1} ]
at the level of subsets. This formulation was incorporated into the period’s developing separation between pointwise function notation and operations induced on families of sets.
Later axiomatic treatments, including those associated with Nicolas Bourbaki, placed the preimage operation within a uniform language of mappings and structures. The resulting notation became common across topology, analysis, probability theory, and abstract algebra.