# Inverse image: Wikis

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# Encyclopedia

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In mathematics, the image of a subset of a function's domain under (or through) the function is the set of all outputs obtained when the function is evaluated at each element of the subset. 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 : XY is a function from set X to 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 AX under f is the subset f[A] ⊆ Y defined by (in set-builder notation):

f[A] = {yY | y = f(x) for some xA}.

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

Let f be a function from X to Y. The preimage or inverse image of a set BY under f is the subset of X defined by

f −1[B] = {xX | 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.

## 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.

## Examples

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: RR 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 = {nR | n < 0} under f is the empty set, because the negative numbers do not have square roots in the set of reals.

3. f: R2R 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 :TMM is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces Tx(M) for xM. This is also an example of a fiber bundle.

## Consequences

Given a function f : XY, 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(A) ⊆ BA ⊆  f −1(B)
• f(f −1(B)) ⊆ B
• f −1(f(A)) ⊇ A
• A1A2f(A1) ⊆ f(A2)
• B1B2f −1(B1) ⊆ f −1(B2)
• f −1(BC) = (f −1(B))C
• (f |A)−1(B) = Af −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).

## Notes

1. ^ Blyth 2005, p. 5

## 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 Creative Commons Attribution/Share-Alike License.

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