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.

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[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
  • f^\rightarrow:\mathcal{P}(X)\rightarrow\mathcal{P}(Y) with f^\rightarrow(A) = \{ f(a)\;|\; a \in A\}
  • f^\leftarrow:\mathcal{P}(Y)\rightarrow\mathcal{P}(X) with f^\leftarrow(B) = \{ a \in X \;|\; f(a) \in B\}.
Star notation
  • f_\star:\mathcal{P}(X)\rightarrow\mathcal{P}(Y) instead of f^\rightarrow.
  • f^\star:\mathcal{P}(Y)\rightarrow\mathcal{P}(X) instead of f^\leftarrow.
Other terminology
  • 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

Boundary of Mandelbrot set as an image of unit circle under map \Psi_M\,
Cardioid as an image of unit circle
Heart curve as an image of unit circle

1. f: {1,2,3} → {a,b,c,d} defined by f(x)=\left\{\begin{matrix} a, & \mbox{if }x=1 \\ a, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \end{matrix}\right.

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 Tx(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(BC) = (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:

  • f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f(A_s)
  • f\left(\bigcap_{s\in S}A_s\right) \subseteq \bigcap_{s\in S} f(A_s)
  • f^{-1}\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f^{-1}(A_s)
  • f^{-1}\left(\bigcap_{s\in S}A_s\right) = \bigcap_{s\in S} f^{-1}(A_s)

(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

This section requires expansion.

[edit] Generalizations

This section requires expansion.

[edit] See also

[edit] Notes

  1. ^ Blyth 2005, p. 5

[edit] References

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

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