Image (mathematics)

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In mathematics, the image of a subset of the domain of a function under the function is the set of all possible outputs obtained when the function is evaluated at each element of the domain. The inverse image or preimage of a particular subset of the codomain of a function is the set of all elements of the domain whose values under the function lie in the chosen subset of the codomain.

Contents 1 Definition of image 2 Definition of inverse image 3 Notation for image and inverse image 4 Examples 5 Consequences 6 Applications 7 Generalizations 8 See also 9 Notes 10 References

[edit] Definition of image

The word "image" is used in three related ways. In these definitions, f : X → Y is a function from set X to set Y.

Image of an element

If x is a member of X, then f(x), the value of f when applied to x, is also called the image of x under f.

Image of a subset

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

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

When there is no risk of confusion, f[A] is simply written as f(A). This convention is a common one; the intended meaning must be inferred from the context. This makes the image of f 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. See Notation below.

Image of a function

The image f[X] of the entire domain X of f, is called simply the image of f .

[edit] Definition of inverse image

Let f be a function from X to Y. 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,denoted by f ⁻¹[{y}] or by f ⁻¹[y], is also called the fiber over y or the level set of y. The set of all the fibers over the elements of Y is a family of sets indexed by Y. This leads to the notion of a fibred category.

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. The two coincide only if f is a bijection.

[edit] Notation for image and inverse image

The traditional notations used in the previous section can be confusing. An alternative^[1] is to give explicit names for the image and preimage as functions between powersets:

Arrow notation

• with
• with .

Star notation

• instead of .

• instead of .

Other terminology

• An alternative notation for f[A] used in mathematical logic and set theory is f "A.

• Some texts refer to the image of f as the range of f, but this usage should be avoided because the word "range" is also commonly used to mean the codomain of f.

[edit] Examples

1. f: {1,2,3} → {a,b,c,d} defined by

The image of the set {2,3} under f is f({2,3}) = {a,c}. The image of the function f is {a,c}. The preimage of a is f ⁻¹({a}) = {1,2}. The preimage of {a,b} is also {1,2}. The preimage of {b,d} is the empty set.

2. f: R → R defined by f(x) = x².

The image of {-2,3} under f is f({-2,3}) = {4,9}, and the image 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.

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. This is also an example of a fiber bundle.

[edit] 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.)

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).

[edit] Applications

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[edit] Generalizations

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[edit] See also

• Range (mathematics)
• Domain (mathematics)
• Bijection, injection and surjection
• Kernel of a function
• Image (category theory)
• Preimage attack (cryptography)

[edit] Notes

[edit] References

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 81-203-0871-9 
• T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, ISBN 1-85233-905-5.

This article incorporates material from Fibre on PlanetMath, which is licensed under the GFDL.