Set of the values of a function
For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.
Inmathematics ,for a function
f
:
X
→
Y
{\displaystyle f:X\to Y}
,theimage of an input value
x
{\displaystyle x}
is the single output value produced by
f
{\displaystyle f}
when passed
x
{\displaystyle x}
.Thepreimage of an output value
y
{\displaystyle y}
is the set of input values that produce
y
{\displaystyle y}
.
More generally, evaluating
f
{\displaystyle f}
at eachelement of a given subset
A
{\displaystyle A}
of itsdomain
X
{\displaystyle X}
produces a set, called the "image of
A
{\displaystyle A}
under (or through)
f
{\displaystyle f}
".Similarly, theinverse image (orpreimage ) of a given subset
B
{\displaystyle B}
of thecodomain
Y
{\displaystyle Y}
is the set of all elements of
X
{\displaystyle X}
that map to a member of
B
.
{\displaystyle B.}
Theimage of the function
f
{\displaystyle f}
is the set of all output values it may produce, that is, the image of
X
{\displaystyle X}
.Thepreimage of
f
{\displaystyle f}
,that is, the preimage of
Y
{\displaystyle Y}
under
f
{\displaystyle f}
,always equals
X
{\displaystyle X}
(thedomain of
f
{\displaystyle f}
); therefore, the former notion is rarely used.
Image and inverse image may also be defined for generalbinary relations ,not just functions.
f
{\displaystyle f}
is a function from domain
X
{\displaystyle X}
to codomain
Y
{\displaystyle Y}
.The image of element
x
{\displaystyle x}
is element
y
{\displaystyle y}
.The preimage of element
y
{\displaystyle y}
is the set {
x
,
x
′
{\displaystyle x,x'}
}. The preimage of element
y
′
{\displaystyle y'}
is
∅
{\displaystyle \varnothing }
.
f
{\displaystyle f}
is a function from domain
X
{\displaystyle X}
to codomain
Y
{\displaystyle Y}
.The image of all elements in subset
A
{\displaystyle A}
is subset
B
{\displaystyle B}
.The preimage of
B
{\displaystyle B}
is subset
C
{\displaystyle C}
f
{\displaystyle f}
is a function from domain
X
{\displaystyle X}
to codomain
Y
.
{\displaystyle Y.}
The yellow oval inside
Y
{\displaystyle Y}
is the image of
f
{\displaystyle f}
.The preimage of
Y
{\displaystyle Y}
is the entire domain
X
{\displaystyle X}
The word "image" is used in three related ways. In these definitions,
f
:
X
→
Y
{\displaystyle f:X\to Y}
is afunction from theset
X
{\displaystyle X}
to the set
Y
.
{\displaystyle Y.}
Image of an element [ edit ]
If
x
{\displaystyle x}
is a member of
X
,
{\displaystyle X,}
then the image of
x
{\displaystyle x}
under
f
,
{\displaystyle f,}
denoted
f
(
x
)
,
{\displaystyle f(x),}
is thevalue of
f
{\displaystyle f}
when applied to
x
.
{\displaystyle x.}
f
(
x
)
{\displaystyle f(x)}
is alternatively known as the output of
f
{\displaystyle f}
for argument
x
.
{\displaystyle x.}
Given
y
,
{\displaystyle y,}
the function
f
{\displaystyle f}
is said totake the value
y
{\displaystyle y}
ortake
y
{\displaystyle y}
as a value if there exists some
x
{\displaystyle x}
in the function's domain such that
f
(
x
)
=
y
.
{\displaystyle f(x)=y.}
Similarly, given a set
S
,
{\displaystyle S,}
f
{\displaystyle f}
is said totake a value in
S
{\displaystyle S}
if there existssome
x
{\displaystyle x}
in the function's domain such that
f
(
x
)
∈
S
.
{\displaystyle f(x)\in S.}
However,
f
{\displaystyle f}
takes [all] values in
S
{\displaystyle S}
and
f
{\displaystyle f}
is valued in
S
{\displaystyle S}
means that
f
(
x
)
∈
S
{\displaystyle f(x)\in S}
forevery point
x
{\displaystyle x}
in the domain of
f
{\displaystyle f}
.
Throughout, let
f
:
X
→
Y
{\displaystyle f:X\to Y}
be a function.
Theimage under
f
{\displaystyle f}
of a subset
A
{\displaystyle A}
of
X
{\displaystyle X}
is the set of all
f
(
a
)
{\displaystyle f(a)}
for
a
∈
A
.
{\displaystyle a\in A.}
It is denoted by
f
[
A
]
,
{\displaystyle f[A],}
or by
f
(
A
)
,
{\displaystyle f(A),}
when there is no risk of confusion. Usingset-builder notation ,this definition can be written as[ 1] [ 2]
f
[
A
]
=
{
f
(
a
)
:
a
∈
A
}
.
{\displaystyle f[A]=\{f(a):a\in A\}.}
This induces a function
f
[
⋅
]
:
P
(
X
)
→
P
(
Y
)
,
{\displaystyle f[\,\cdot \,]:{\mathcal {P}}(X)\to {\mathcal {P}}(Y),}
where
P
(
S
)
{\displaystyle {\mathcal {P}}(S)}
denotes thepower set of a set
S
;
{\displaystyle S;}
that is the set of allsubsets of
S
.
{\displaystyle S.}
See§ Notation below for more.
Image of a function [ edit ]
Theimage of a function is the image of its entiredomain ,also known as therange of the function.[ 3] This last usage should be avoided because the word "range" is also commonly used to mean thecodomain of
f
.
{\displaystyle f.}
Generalization to binary relations [ edit ]
If
R
{\displaystyle R}
is an arbitrarybinary relation on
X
×
Y
,
{\displaystyle X\times Y,}
then the set
{
y
∈
Y
:
x
R
y
for some
x
∈
X
}
{\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}}
is called the image, or the range, of
R
.
{\displaystyle R.}
Dually, the set
{
x
∈
X
:
x
R
y
for some
y
∈
Y
}
{\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}}
is called the domain of
R
.
{\displaystyle R.}
"Preimage" redirects here. For the cryptographic attack on hash functions, see
preimage attack .
Let
f
{\displaystyle f}
be a function from
X
{\displaystyle X}
to
Y
.
{\displaystyle Y.}
Thepreimage orinverse image of a set
B
⊆
Y
{\displaystyle B\subseteq Y}
under
f
,
{\displaystyle f,}
denoted by
f
−
1
[
B
]
,
{\displaystyle f^{-1}[B],}
is the subset of
X
{\displaystyle X}
defined by
f
−
1
[
B
]
=
{
x
∈
X
:
f
(
x
)
∈
B
}
.
{\displaystyle f^{-1}[B]=\{x\in X\,:\,f(x)\in B\}.}
Other notations include
f
−
1
(
B
)
{\displaystyle f^{-1}(B)}
and
f
−
(
B
)
.
{\displaystyle f^{-}(B).}
The inverse image of asingleton set ,denoted by
f
−
1
[
{
y
}
]
{\displaystyle f^{-1}[\{y\}]}
or by
f
−
1
[
y
]
,
{\displaystyle f^{-1}[y],}
is also called thefiber or fiber over
y
{\displaystyle y}
or thelevel set of
y
.
{\displaystyle y.}
The set of all the fibers over the elements of
Y
{\displaystyle Y}
is a family of sets indexed by
Y
.
{\displaystyle Y.}
For example, for the function
f
(
x
)
=
x
2
,
{\displaystyle f(x)=x^{2},}
the inverse image of
{
4
}
{\displaystyle \{4\}}
would be
{
−
2
,
2
}
.
{\displaystyle \{-2,2\}.}
Again, if there is no risk of confusion,
f
−
1
[
B
]
{\displaystyle f^{-1}[B]}
can be denoted by
f
−
1
(
B
)
,
{\displaystyle f^{-1}(B),}
and
f
−
1
{\displaystyle f^{-1}}
can also be thought of as a function from the power set of
Y
{\displaystyle Y}
to the power set of
X
.
{\displaystyle X.}
The notation
f
−
1
{\displaystyle f^{-1}}
should not be confused with that forinverse function ,although it coincides with the usual one for bijections in that the inverse image of
B
{\displaystyle B}
under
f
{\displaystyle f}
is the image of
B
{\displaystyle B}
under
f
−
1
.
{\displaystyle f^{-1}.}
Notation for image and inverse image[ edit ]
The traditional notations used in the previous section do not distinguish the original function
f
:
X
→
Y
{\displaystyle f:X\to Y}
from the image-of-sets function
f
:
P
(
X
)
→
P
(
Y
)
{\displaystyle f:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)}
;likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). Given the right context, this keeps the notation light and usually does not cause confusion. But if needed, an alternativeis to give explicit names for the image and preimage as functions between power sets:
f
→
:
P
(
X
)
→
P
(
Y
)
{\displaystyle f^{\rightarrow }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)}
with
f
→
(
A
)
=
{
f
(
a
)
|
a
∈
A
}
{\displaystyle f^{\rightarrow }(A)=\{f(a)\;|\;a\in A\}}
f
←
:
P
(
Y
)
→
P
(
X
)
{\displaystyle f^{\leftarrow }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)}
with
f
←
(
B
)
=
{
a
∈
X
|
f
(
a
)
∈
B
}
{\displaystyle f^{\leftarrow }(B)=\{a\in X\;|\;f(a)\in B\}}
f
⋆
:
P
(
X
)
→
P
(
Y
)
{\displaystyle f_{\star }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)}
instead of
f
→
{\displaystyle f^{\rightarrow }}
f
⋆
:
P
(
Y
)
→
P
(
X
)
{\displaystyle f^{\star }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)}
instead of
f
←
{\displaystyle f^{\leftarrow }}
An alternative notation for
f
[
A
]
{\displaystyle f[A]}
used inmathematical logic andset theory is
f
″
A
.
{\displaystyle f\,''A.}
[ 6] [ 7]
Some texts refer to the image of
f
{\displaystyle f}
as the range of
f
,
{\displaystyle f,}
[ 8] but this usage should be avoided because the word "range" is also commonly used to mean thecodomain of
f
.
{\displaystyle f.}
f
:
{
1
,
2
,
3
}
→
{
a
,
b
,
c
,
d
}
{\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}}
defined by
{
1
↦
a
,
2
↦
a
,
3
↦
c
.
{\displaystyle \left\{{\begin{matrix}1\mapsto a,\\2\mapsto a,\\3\mapsto c.\end{matrix}}\right.}
Theimage of the set
{
2
,
3
}
{\displaystyle \{2,3\}}
under
f
{\displaystyle f}
is
f
(
{
2
,
3
}
)
=
{
a
,
c
}
.
{\displaystyle f(\{2,3\})=\{a,c\}.}
Theimage of the function
f
{\displaystyle f}
is
{
a
,
c
}
.
{\displaystyle \{a,c\}.}
Thepreimage of
a
{\displaystyle a}
is
f
−
1
(
{
a
}
)
=
{
1
,
2
}
.
{\displaystyle f^{-1}(\{a\})=\{1,2\}.}
Thepreimage of
{
a
,
b
}
{\displaystyle \{a,b\}}
is also
f
−
1
(
{
a
,
b
}
)
=
{
1
,
2
}
.
{\displaystyle f^{-1}(\{a,b\})=\{1,2\}.}
Thepreimage of
{
b
,
d
}
{\displaystyle \{b,d\}}
under
f
{\displaystyle f}
is theempty set
{
}
=
∅
.
{\displaystyle \{\ \}=\emptyset.}
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
defined by
f
(
x
)
=
x
2
.
{\displaystyle f(x)=x^{2}.}
Theimage of
{
−
2
,
3
}
{\displaystyle \{-2,3\}}
under
f
{\displaystyle f}
is
f
(
{
−
2
,
3
}
)
=
{
4
,
9
}
,
{\displaystyle f(\{-2,3\})=\{4,9\},}
and theimage of
f
{\displaystyle f}
is
R
+
{\displaystyle \mathbb {R} ^{+}}
(the set of all positive real numbers and zero). Thepreimage of
{
4
,
9
}
{\displaystyle \{4,9\}}
under
f
{\displaystyle f}
is
f
−
1
(
{
4
,
9
}
)
=
{
−
3
,
−
2
,
2
,
3
}
.
{\displaystyle f^{-1}(\{4,9\})=\{-3,-2,2,3\}.}
Thepreimage of set
N
=
{
n
∈
R
:
n
<
0
}
{\displaystyle N=\{n\in \mathbb {R}:n<0\}}
under
f
{\displaystyle f}
is the empty set, because the negative numbers do not have square roots in the set of reals.
f
:
R
2
→
R
{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }
defined by
f
(
x
,
y
)
=
x
2
+
y
2
.
{\displaystyle f(x,y)=x^{2}+y^{2}.}
Thefibers
f
−
1
(
{
a
}
)
{\displaystyle f^{-1}(\{a\})}
areconcentric circles about theorigin ,the origin itself, and theempty set (respectively), depending on whether
a
>
0
,
a
=
0
,
or
a
<
0
{\displaystyle a>0,\ a=0,{\text{ or }}\ a<0}
(respectively). (If
a
≥
0
,
{\displaystyle a\geq 0,}
then thefiber
f
−
1
(
{
a
}
)
{\displaystyle f^{-1}(\{a\})}
is the set of all
(
x
,
y
)
∈
R
2
{\displaystyle (x,y)\in \mathbb {R} ^{2}}
satisfying the equation
x
2
+
y
2
=
a
,
{\displaystyle x^{2}+y^{2}=a,}
that is, the origin-centered circle with radius
a
.
{\displaystyle {\sqrt {a}}.}
)
If
M
{\displaystyle M}
is amanifold and
π
:
T
M
→
M
{\displaystyle \pi:TM\to M}
is the canonicalprojection from thetangent bundle
T
M
{\displaystyle TM}
to
M
,
{\displaystyle M,}
then thefibers of
π
{\displaystyle \pi }
are thetangent spaces
T
x
(
M
)
for
x
∈
M
.
{\displaystyle T_{x}(M){\text{ for }}x\in M.}
This is also an example of afiber bundle .
Aquotient group is a homomorphicimage .
Counter-examples based on thereal numbers
R
,
{\displaystyle \mathbb {R},}
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
defined by
x
↦
x
2
,
{\displaystyle x\mapsto x^{2},}
showing that equality generally need not hold for some laws:
Image showing non-equal sets:
f
(
A
∩
B
)
⊊
f
(
A
)
∩
f
(
B
)
.
{\displaystyle f\left(A\cap B\right)\subsetneq f(A)\cap f(B).}
The sets
A
=
[
−
4
,
2
]
{\displaystyle A=[-4,2]}
and
B
=
[
−
2
,
4
]
{\displaystyle B=[-2,4]}
are shown inblue immediately below the
x
{\displaystyle x}
-axis while their intersection
A
3
=
[
−
2
,
2
]
{\displaystyle A_{3}=[-2,2]}
is shown ingreen .
f
(
f
−
1
(
B
3
)
)
⊊
B
3
.
{\displaystyle f\left(f^{-1}\left(B_{3}\right)\right)\subsetneq B_{3}.}
f
−
1
(
f
(
A
4
)
)
⊋
A
4
.
{\displaystyle f^{-1}\left(f\left(A_{4}\right)\right)\supsetneq A_{4}.}
For every function
f
:
X
→
Y
{\displaystyle f:X\to Y}
and all subsets
A
⊆
X
{\displaystyle A\subseteq X}
and
B
⊆
Y
,
{\displaystyle B\subseteq Y,}
the following properties hold:
Image
Preimage
f
(
X
)
⊆
Y
{\displaystyle f(X)\subseteq Y}
f
−
1
(
Y
)
=
X
{\displaystyle f^{-1}(Y)=X}
f
(
f
−
1
(
Y
)
)
=
f
(
X
)
{\displaystyle f\left(f^{-1}(Y)\right)=f(X)}
f
−
1
(
f
(
X
)
)
=
X
{\displaystyle f^{-1}(f(X))=X}
f
(
f
−
1
(
B
)
)
⊆
B
{\displaystyle f\left(f^{-1}(B)\right)\subseteq B}
(equal if
B
⊆
f
(
X
)
;
{\displaystyle B\subseteq f(X);}
for instance, if
f
{\displaystyle f}
is surjective)[ 9] [ 10]
f
−
1
(
f
(
A
)
)
⊇
A
{\displaystyle f^{-1}(f(A))\supseteq A}
(equal if
f
{\displaystyle f}
is injective)[ 9] [ 10]
f
(
f
−
1
(
B
)
)
=
B
∩
f
(
X
)
{\displaystyle f(f^{-1}(B))=B\cap f(X)}
(
f
|
A
)
−
1
(
B
)
=
A
∩
f
−
1
(
B
)
{\displaystyle \left(f\vert _{A}\right)^{-1}(B)=A\cap f^{-1}(B)}
f
(
f
−
1
(
f
(
A
)
)
)
=
f
(
A
)
{\displaystyle f\left(f^{-1}(f(A))\right)=f(A)}
f
−
1
(
f
(
f
−
1
(
B
)
)
)
=
f
−
1
(
B
)
{\displaystyle f^{-1}\left(f\left(f^{-1}(B)\right)\right)=f^{-1}(B)}
f
(
A
)
=
∅
if and only if
A
=
∅
{\displaystyle f(A)=\varnothing \,{\text{ if and only if }}\,A=\varnothing }
f
−
1
(
B
)
=
∅
if and only if
B
⊆
Y
∖
f
(
X
)
{\displaystyle f^{-1}(B)=\varnothing \,{\text{ if and only if }}\,B\subseteq Y\setminus f(X)}
f
(
A
)
⊇
B
if and only if
there exists
C
⊆
A
such that
f
(
C
)
=
B
{\displaystyle f(A)\supseteq B\,{\text{ if and only if }}{\text{ there exists }}C\subseteq A{\text{ such that }}f(C)=B}
f
−
1
(
B
)
⊇
A
if and only if
f
(
A
)
⊆
B
{\displaystyle f^{-1}(B)\supseteq A\,{\text{ if and only if }}\,f(A)\subseteq B}
f
(
A
)
⊇
f
(
X
∖
A
)
if and only if
f
(
A
)
=
f
(
X
)
{\displaystyle f(A)\supseteq f(X\setminus A)\,{\text{ if and only if }}\,f(A)=f(X)}
f
−
1
(
B
)
⊇
f
−
1
(
Y
∖
B
)
if and only if
f
−
1
(
B
)
=
X
{\displaystyle f^{-1}(B)\supseteq f^{-1}(Y\setminus B)\,{\text{ if and only if }}\,f^{-1}(B)=X}
f
(
X
∖
A
)
⊇
f
(
X
)
∖
f
(
A
)
{\displaystyle f(X\setminus A)\supseteq f(X)\setminus f(A)}
f
−
1
(
Y
∖
B
)
=
X
∖
f
−
1
(
B
)
{\displaystyle f^{-1}(Y\setminus B)=X\setminus f^{-1}(B)}
[ 9]
f
(
A
∪
f
−
1
(
B
)
)
⊆
f
(
A
)
∪
B
{\displaystyle f\left(A\cup f^{-1}(B)\right)\subseteq f(A)\cup B}
[ 11]
f
−
1
(
f
(
A
)
∪
B
)
⊇
A
∪
f
−
1
(
B
)
{\displaystyle f^{-1}(f(A)\cup B)\supseteq A\cup f^{-1}(B)}
[ 11]
f
(
A
∩
f
−
1
(
B
)
)
=
f
(
A
)
∩
B
{\displaystyle f\left(A\cap f^{-1}(B)\right)=f(A)\cap B}
[ 11]
f
−
1
(
f
(
A
)
∩
B
)
⊇
A
∩
f
−
1
(
B
)
{\displaystyle f^{-1}(f(A)\cap B)\supseteq A\cap f^{-1}(B)}
[ 11]
Also:
f
(
A
)
∩
B
=
∅
if and only if
A
∩
f
−
1
(
B
)
=
∅
{\displaystyle f(A)\cap B=\varnothing \,{\text{ if and only if }}\,A\cap f^{-1}(B)=\varnothing }
For functions
f
:
X
→
Y
{\displaystyle f:X\to Y}
and
g
:
Y
→
Z
{\displaystyle g:Y\to Z}
with subsets
A
⊆
X
{\displaystyle A\subseteq X}
and
C
⊆
Z
,
{\displaystyle C\subseteq Z,}
the following properties hold:
(
g
∘
f
)
(
A
)
=
g
(
f
(
A
)
)
{\displaystyle (g\circ f)(A)=g(f(A))}
(
g
∘
f
)
−
1
(
C
)
=
f
−
1
(
g
−
1
(
C
)
)
{\displaystyle (g\circ f)^{-1}(C)=f^{-1}(g^{-1}(C))}
Multiple subsets of domain or codomain [ edit ]
For function
f
:
X
→
Y
{\displaystyle f:X\to Y}
and subsets
A
,
B
⊆
X
{\displaystyle A,B\subseteq X}
and
S
,
T
⊆
Y
,
{\displaystyle S,T\subseteq Y,}
the following properties hold:
Image
Preimage
A
⊆
B
implies
f
(
A
)
⊆
f
(
B
)
{\displaystyle A\subseteq B\,{\text{ implies }}\,f(A)\subseteq f(B)}
S
⊆
T
implies
f
−
1
(
S
)
⊆
f
−
1
(
T
)
{\displaystyle S\subseteq T\,{\text{ implies }}\,f^{-1}(S)\subseteq f^{-1}(T)}
f
(
A
∪
B
)
=
f
(
A
)
∪
f
(
B
)
{\displaystyle f(A\cup B)=f(A)\cup f(B)}
[ 11] [ 12]
f
−
1
(
S
∪
T
)
=
f
−
1
(
S
)
∪
f
−
1
(
T
)
{\displaystyle f^{-1}(S\cup T)=f^{-1}(S)\cup f^{-1}(T)}
f
(
A
∩
B
)
⊆
f
(
A
)
∩
f
(
B
)
{\displaystyle f(A\cap B)\subseteq f(A)\cap f(B)}
[ 11] [ 12] (equal if
f
{\displaystyle f}
is injective[ 13] )
f
−
1
(
S
∩
T
)
=
f
−
1
(
S
)
∩
f
−
1
(
T
)
{\displaystyle f^{-1}(S\cap T)=f^{-1}(S)\cap f^{-1}(T)}
f
(
A
∖
B
)
⊇
f
(
A
)
∖
f
(
B
)
{\displaystyle f(A\setminus B)\supseteq f(A)\setminus f(B)}
[ 11] (equal if
f
{\displaystyle f}
is injective[ 13] )
f
−
1
(
S
∖
T
)
=
f
−
1
(
S
)
∖
f
−
1
(
T
)
{\displaystyle f^{-1}(S\setminus T)=f^{-1}(S)\setminus f^{-1}(T)}
[ 11]
f
(
A
△
B
)
⊇
f
(
A
)
△
f
(
B
)
{\displaystyle f\left(A\triangle B\right)\supseteq f(A)\triangle f(B)}
(equal if
f
{\displaystyle f}
is injective)
f
−
1
(
S
△
T
)
=
f
−
1
(
S
)
△
f
−
1
(
T
)
{\displaystyle f^{-1}\left(S\triangle T\right)=f^{-1}(S)\triangle f^{-1}(T)}
The results relating images and preimages to the (Boolean ) algebra ofintersection andunion work for any collection of subsets, not just for pairs of subsets:
f
(
⋃
s
∈
S
A
s
)
=
⋃
s
∈
S
f
(
A
s
)
{\displaystyle f\left(\bigcup _{s\in S}A_{s}\right)=\bigcup _{s\in S}f\left(A_{s}\right)}
f
(
⋂
s
∈
S
A
s
)
⊆
⋂
s
∈
S
f
(
A
s
)
{\displaystyle f\left(\bigcap _{s\in S}A_{s}\right)\subseteq \bigcap _{s\in S}f\left(A_{s}\right)}
f
−
1
(
⋃
s
∈
S
B
s
)
=
⋃
s
∈
S
f
−
1
(
B
s
)
{\displaystyle f^{-1}\left(\bigcup _{s\in S}B_{s}\right)=\bigcup _{s\in S}f^{-1}\left(B_{s}\right)}
f
−
1
(
⋂
s
∈
S
B
s
)
=
⋂
s
∈
S
f
−
1
(
B
s
)
{\displaystyle f^{-1}\left(\bigcap _{s\in S}B_{s}\right)=\bigcap _{s\in S}f^{-1}\left(B_{s}\right)}
(Here,
S
{\displaystyle S}
can be infinite, evenuncountably infinite .)
With respect to the algebra of subsets described above, the inverse image function is alattice homomorphism ,while the image function is only asemilattice homomorphism (that is, it does not always preserve intersections).
^ "5.4: Onto Functions and Images/Preimages of Sets" .Mathematics LibreTexts .2019-11-05.Retrieved2020-08-28 .
^ Paul R. Halmos (1968).Naive Set Theory .Princeton: Nostrand. Here: Sect.8
^ Weisstein, Eric W."Image" .mathworld.wolfram .Retrieved2020-08-28 .
^ Jean E. Rubin (1967).Set Theory for the Mathematician .Holden-Day. p. xix.ASIN B0006BQH7S .
^ M. Randall Holmes:Inhomogeneity of the urelements in the usual models of NFU ,December 29, 2005, on: Semantic Scholar, p. 2
^ Hoffman, Kenneth (1971).Linear Algebra (2nd ed.). Prentice-Hall. p. 388.
^a b c SeeHalmos 1960 ,p. 31
^a b SeeMunkres 2000 ,p. 19
^a b c d e f g h See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.
^a b Kelley 1985 ,p.85
^a b SeeMunkres 2000 ,p. 21
This article incorporates material from Fibre onPlanetMath ,which is licensed under theCreative Commons Attribution/Share-Alike License .