In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. The image of a subset A ⊆ X under f is the subset of Y defined by Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

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

Notice that the range of f is the image f(X) of its domain X. In mathematics, the range of a function is the set of all values produced by a function. ... In mathematics, the domain of a function is the set of all input values to the function. ...

With this definition, the image f becomes a function whose domain is the set of all subsets of X (also known as the power set of X) and whose codomain is the power set of Y. Note that the same notation is used for the original function f and its image. This is a common convention; the intended usage must be inferred by context. In mathematics, a set S, the power set of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...

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 called a fiber or fibre, or level set. In mathematics, a singleton is a set with exactly one element. ... In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ...

Note that f ⁻¹ should not be confused with the inverse function. The two only coincide if f is bijective. f ⁻¹ is a new function whose domain is the power set of Y and whose codomain is the power set of X. In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

Looking at it the other way, f can be seen as a family of sets indexed by Y. An obvious extension of this idea is that of fibred category. In mathematics, specifically in category theory, a fibred category or fibered category, is a functor such that every morphism in has a unique cartesian lift. ...

Examples

1. f: {1,2,3} → {a,b,c,d} defined by

In this example, 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.

In this example, the image of {-2,3} under f is f({-2,3})={4,9} and the range of f is the set of nonnegative real numbers. The preimage of {4,9} under f is f ⁻¹({4,9})={-2,2,-3,3}.

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

In this example, the fibres f ⁻¹({a}), are concentric circles about the origin (mathematics), the origin, and the empty set, depending on whether a is > 0, a = 0, or a < 0 respectively. Concentric objects share the same center or origin. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ... In mathematics, the empty set is the set with no elements. ...

4. If M is a manifold and This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...

is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces The word projection can mean more than one thing. ... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...

T_x(M) for . This is also an example of a fiber bundle.

In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...

Consequences

Some consequences that follow immediately from these definitions are:

• 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 implies f(A₁) ⊆ f(A₂)
• B₁ ⊆ B₂ implies f ⁻¹(B₁) ⊆ f ⁻¹(B₂)
• f ⁻¹(B^C) = (f ⁻¹(B))^C
• (f |_A)⁻¹(B) = A ∩ f ⁻¹(B)

These are valid for arbitrary subsets A, A₁ and A₂ of the domain and arbitrary subsets B, B₁ and B₂ of the codomain. The results relating images and preimages to the algebra of intersection and union work for any collections of subsets, not just for pairs of subsets. 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, all elements of B that also belong to A), but no other elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

See also

• Preimage_attack (cryptography)
• Image (category theory)
• Kernel of a function

This article incorporates material from Fibre on PlanetMath, which is licensed under the GFDL. In cryptography, a preimage attack on a cryptographic hash differs from a collision attack. ... Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property: There exists a morphism such that f = hg. ... In mathematics, the kernel of a function f may be taken to be either the equivalence relation on the functions domain that roughly expresses the idea of equivalent as far as the function f can tell, or the corresponding partition of the domain. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...