image (mathematics)
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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 :
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) =
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 ∈ 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 .
Inverse image
{{Redirect|Preimage|the cryptographic attack on hash functions|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 ∈ 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.
Notation for image and inverse image">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
- Star notation
-
f*:Scri(tP(X)→Scri(tP(Y)
instead of f→
.
-
f*:Scri(tP(Y)→Scri(tP(X)
instead of f←
.
- 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
File:Jung200.png|right|thumb|Boundary of
Mandelbrot set as an image of
unit circle under (external ray|map
ΨM
) ]]File:Circle2cardioid.png|right|thumb|
Cardioid as an image of
unit circleunit circleFile:Circle2heart.png|right|thumb|Heart curve as an image of
unit circleunit circle1.
f: {1,2,3} → {
a,b,c,d} defined by
f(x)=(matrix a & if x=1 a & if x=2 c & if x=3. endmatrix&nbs(;).
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 Tx(
M) for
x∈
M. This is also an example of a
fiber bundle.
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(A) ⊆ B ⇔ A ⊆ f −1(B)
- 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((big&cu(;s∈ SAs&nbs(;)) = big&cu(;s∈ S f(As)
f((big&ca(;s∈ SAs&nbs(;)) &su(;eq big&ca(;s∈ S f(As)
f-1((big&cu(;s∈ SAs&nbs(;)) = big&cu(;s∈ S f-1(As)
f-1((big&ca(;s∈ SAs&nbs(;)) = big&ca(;s∈ S f-1(As)
(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).
See also
Notes
-
[Blyth 2005, p. 5]
References
- {{Citation |authorlink=Michael Artin | last= Artin | first= Michael | title= Algebra | edition=| year=1991 | publisher=Prentice Hall| isbn= 81-203-0871-9}}
- T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, ISBN 1-85233-905-5.
{{planetmath|id=3276|title=Fibre}}
Imatge (matemàtiques)Obor hodnotVærdimængdeBild (Mathematik)Image (mathématiques)Immagine (matematica)Beeld (wiskunde)Obraz (matematyka)Imagem (matemática)Värdemängdவீச்சு, எதிருரு மற்றும் முன்னுரு
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