Image (mathematics)
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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
- 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.
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.
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.
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.
Looking at it the other way, <math>f<math> can be seen as a family of sets indexed by <math>Y<math>. An obvious extension of this idea is that of fibred category.
Examples
1. f: {1,2,3} → {a,b,c,d} defined by <math>f(x)=\left\{\begin{matrix} a, & \mbox{if }x=1 \\ d, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \end{matrix}\right.<math>
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 + x2.
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.
4. If M is a manifold and
- <math>\pi\colon TM\to M<math>
is the canonical projection from the tangent bundle TM to M, then the fibres of <math>\pi<math> are the tangent spaces
- <math>T_x(M)<math> for <math>x\in M<math>. This is also an example of a fiber bundle.
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)
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.