Image (mathematics)

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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.

Contents 1 Definition 2 Uniform arrow notations 3 Examples 4 Consequences 5 See also 6 Notes 7 References

[edit] Definition

If f : X → Y is a function from set X to set Y and x is a member of X, then f(x), the image of x under f, is a unique member of Y that f associates with x. The image under f of the entire domain X is often called the range of f, and is a subset of the codomain Y.

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 simply written as f(A). An alternative notation for f[A] that is common in the older literature mathematical logic and still preferred by some set theorists, is f "A.

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. 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.

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.

f can also be seen as a family of sets indexed by Y, which leads to the notion of a fibred category.

[edit] Uniform arrow notations

The traditional notations used in the previous section can be confusing. An alternative^[1] is to explicitly write the image and preimage as two functions in their own right: with and with . If we consider the powerset as a poset ordered by inclusion, then the image and preimage functions are monotone.

[edit] 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.

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] 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.