Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function. The inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S.

Image and inverse image may also be defined for general binary relations, not just functions.

Definition

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

Image of an element

If x is a member of X, then f(x) = y (the value of f when applied to x) is the image of x under f. y is alternatively known as the output of f for argument x.

Image of a subset

The image of a subset A ⊆ X under f is the subset f[A] ⊆ Y defined by (in set-builder notation):

f[A] = \{ \, y \in Y \, | \, y = f(x) \text{ for some } x \in 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.

Inverse image

"Preimage" redirects here. For the cryptographic attack on hash functions, see preimage attack.

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 \in X \, | \, f(x) \in 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.

For example, for the function f(x) = x², the inverse image of {4} would be {-2,2}. 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. In fact, the two are never the same, even if f is a bijection.

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

An alternative notation for f[A] used in mathematical logic and set theory is f "A.^[2]
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.

Examples

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}. The preimage of set N = {n ∈ R | n < 0} under f is the empty set, because the negative numbers do not have square roots in the set of reals.

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

The fibres f⁻¹({a}) are concentric circles about the origin, the origin itself, 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.

5. A quotient group is a Homomorphic image

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)^[3]
f(A₁ ∩ A₂) ⊆ f(A₁) ∩ f(A₂)^[3]
f⁻¹(B₁ ∪ B₂) = f⁻¹(B₁) ∪ f⁻¹(B₂)
f⁻¹(B₁ ∩ B₂) = f⁻¹(B₁) ∩ f⁻¹(B₂)
f(A) ⊆ B ⇔ A ⊆ f⁻¹(B)
f(f⁻¹(B)) ⊆ B^[4]
f⁻¹(f(A)) ⊇ A^[5]
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:

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

(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).

Notes

↑ Blyth 2005, p. 5
↑ Jean E. Rubin (1967). Set Theory for the Mathematician. Holden-Day. p. xix. ASIN B0006BQH7S.
1 2 Kelley (1985), p. 85
↑ Equality holds if B is a subset of Im(f) or, in particular, if f is surjective. See Munkres, J.. Topology (2000), p. 19.
↑ Equality holds if f is injective. See Munkres, J.. Topology (2000), p. 19.

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.
Munkres, James R. (2000). Topology (2 ed.). Prentice Hall. ISBN 978-0-13-181629-9.
Kelley, John L. (1985). General Topology. Graduate Texts in Mathematics 27 (2 ed.). Birkhäuser. ISBN 978-0-387-90125-1.

This article incorporates material from Fibre on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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