In mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage, successively, as the function's argument. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

Definition

Let X and Y be sets, f be the function f : X → Y, and x be some member of X. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Then the image of x under f, denoted f(x), is the unique member y of Y that f associates with x. The image of a function f is denoted im(f) and is the range of f, or more precisely, the image of its domain. In Mathematics, the range of a function is the set of all "output" values produced by that function In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined

The image of a subset A ⊆ X under f is the subset of Y defined by

f[A] = {y ∈ Y | y = f(x) for some x ∈ A}.

When there is no risk of confusion, f[A] is sometimes simply written f(A). An alternative notation for f[A], common in the older literature on mathematical logic and still preferred by some set theorists, is f "A. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic.

Given this definition, the image of f becomes a function whose domain is the power set of X (the set of all subsets of X), and whose codomain is the power set of Y. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Mathematics, the codomain, or target, of a function f: X → Y is the set The same notation can denote either the function f or its image. This convention is a common one; the intended meaning must be inferred from the context.

The preimage or inverse image of a set B ⊆ Y under f is the subset of X defined by

f ⁻¹[B] = {x ∈ X | f(x) ∈ B}.

The inverse image of a singleton, f ⁻¹[{y}], is a fiber (also spelled fibre) or a level set. In Mathematics, a singleton is a set with exactly one element In Mathematics, the fiber of a point y under a function f    X  →  Y is the inverse In Mathematics, a level set of a real -valued function f of n variables is a set of the form { ( x 1

Again, if there is no risk of confusion, we may denote f ⁻¹[B] by f ⁻¹(B), and think of f ⁻¹ as a function from the power set of Y to the power set of X. The notation f ⁻¹ should not be confused with that for inverse function. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B The two coincide only if f is a bijection. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

f can also be seen as a family of sets indexed by Y, which leads to the notion of a fibred category. Fibred categories are abstract entities in Mathematics used to provide a general framework for Descent theory.

Examples

1. f: {1,2,3} → {a,b,c,d} defined by

The image of {2,3} under f is f({2,3}) = {d,c}, and the range of f is {a,d,c}. The preimage of {a,c} is f ⁻¹({a,c}) = {1,3}.

2. f: R → R defined by f(x) = x².

The image of {-2,3} under f is f({-2,3}) = {4,9}, and the range of f is R⁺. The preimage of {4,9} under f is f ⁻¹({4,9}) = {-3,-2,2,3}.

3. f: R² → R defined by f(x, y) = x² + y².

The fibres f ⁻¹({a}) are concentric circles about the origin, the origin, and the empty set, depending on whether a>0, a=0, or a<0, respectively. Concentric objects share the same center, axis or origin with one inside the other In Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

4. If M is a manifold and π :TM→M is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces T_x(M) for x∈M. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since This is also an example of a fiber bundle. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space.

Consequences

Given a function f : X → Y, for all subsets A, A₁, and A₂ of X and all subsets B, B₁, and B₂ of Y we have:

• f(A₁ ∪ A₂) = f(A₁) ∪ f(A₂)
• f(A₁ ∩ A₂) ⊆ f(A₁) ∩ f(A₂)
• f ⁻¹(B₁ ∪ B₂) = f ⁻¹(B₁) ∪ f ⁻¹(B₂)
• f ⁻¹(B₁ ∩ B₂) = f ⁻¹(B₁) ∩ f ⁻¹(B₂)
• f(f ⁻¹(B)) ⊆ B
• f ⁻¹(f(A)) ⊇ A
• A₁ ⊆ A₂ → f(A₁) ⊆ f(A₂)
• B₁ ⊆ B₂ → f ⁻¹(B₁) ⊆ f ⁻¹(B₂)
• f ⁻¹(B^C) = (f ⁻¹(B))^C
• (f |_A)⁻¹(B) = A ∩ f ⁻¹(B).

The results relating images and preimages to the (Boolean) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets:

(here S can be infinite, even uncountably infinite. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets )

With respect to the algebra of subsets, by the above we see that the inverse image function is a lattice homomorphism while the image function is only a semilattice homomorphism (it does not always preserve intersections). In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' A semilattice is a mathematical concept with two definitions one as a type of Ordered set, the other as an Algebraic structure.

See also

• range (mathematics)
• domain (mathematics)
• function (mathematics)
• bijection, injection and surjection
• kernel of a function
• image (category theory)
• preimage attack (cryptography)

References

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 81-203-0871-9 

This article incorporates material from Fibre on PlanetMath, which is licensed under the GFDL. In Mathematics, the range of a function is the set of all "output" values produced by that function In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments In Mathematics, the kernel of a function f may be taken to be either the Equivalence relation on the function's domain Given a category C and a Morphism fX\rightarrow Y in C, the image of f is a Monomorphism hI\rightarrow In Cryptography, a preimage attack on a cryptographic hash is an attempt to find a message that has a specific hash value Michael Artin (born 1934 is an American Mathematician and a professor at MIT, known for his contributions to Algebraic PlanetMath is a free, collaborative online Mathematics Encyclopedia.