In

Inhomogeneity of the urelements in the usual models of NFU

December 29, 2005, on: Semantic Scholar, p. 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

$f\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $x\; \backslash mapsto\; x^2,$

showing that equality generally need

not hold for some laws: , - , , - , center, upright=1.7, $f\backslash left(f^\{-1\}\backslash left(B\_3\backslash right)\backslash right)\; \backslash subsetneq\; B\_3.$ , - ,

(equal if $B\; \backslash subseteq\; f(X);$ for instance, if $f$ is surjective)See See , $f^\{-1\}(f(A))\; \backslash supseteq\; A$

(equal if $f$ is injective) , - , $f(f^\{-1\}(B))\; =\; B\; \backslash cap\; f(X)$ , $\backslash left(f\; \backslash vert\_A\backslash right)^\{-1\}(B)\; =\; A\; \backslash cap\; f^\{-1\}(B)$ , - , $f\backslash left(f^\{-1\}(f(A))\backslash right)\; =\; f(A)$ , $f^\{-1\}\backslash left(f\backslash left(f^\{-1\}(B)\backslash right)\backslash right)\; =\; f^\{-1\}(B)$ , - , $f(A)\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; A\; =\; \backslash varnothing$ , $f^\{-1\}(B)\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; B\; \backslash subseteq\; Y\; \backslash setminus\; f(X)$ , - , $f(A)\; \backslash supseteq\; B\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\; \backslash text\{\; there\; exists\; \}\; C\; \backslash subseteq\; A\; \backslash text\{\; such\; that\; \}\; f(C)\; =\; B$ , $f^\{-1\}(B)\; \backslash supseteq\; A\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f(A)\; \backslash subseteq\; B$ , - , $f(A)\; \backslash supseteq\; f(X\; \backslash setminus\; A)\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f(A)\; =\; f(X)$ , $f^\{-1\}(B)\; \backslash supseteq\; f^\{-1\}(Y\; \backslash setminus\; B)\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f^\{-1\}(B)\; =\; X$ , - , $f(X\; \backslash setminus\; A)\; \backslash supseteq\; f(X)\; \backslash setminus\; f(A)$ , $f^\{-1\}(Y\; \backslash setminus\; B)\; =\; X\; \backslash setminus\; f^\{-1\}(B)$ , - , $f\backslash left(A\; \backslash cup\; f^\{-1\}(B)\backslash right)\; \backslash subseteq\; f(A)\; \backslash cup\; B$See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. , $f^\{-1\}(f(A)\; \backslash cup\; B)\; \backslash supseteq\; A\; \backslash cup\; f^\{-1\}(B)$ , - , $f\backslash left(A\; \backslash cap\; f^\{-1\}(B)\backslash right)\; =\; f(A)\; \backslash cap\; B$ , $f^\{-1\}(f(A)\; \backslash cap\; B)\; \backslash supseteq\; A\; \backslash cap\; f^\{-1\}(B)$ Also: * $f(A)\; \backslash cap\; B\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; A\; \backslash cap\; f^\{-1\}(B)\; =\; \backslash varnothing$

(equal if $f$ is injectiveSee ) , $f^\{-1\}(S\; \backslash cap\; T)\; =\; f^\{-1\}(S)\; \backslash cap\; f^\{-1\}(T)$ , - , $f(A\; \backslash setminus\; B)\; \backslash supseteq\; f(A)\; \backslash setminus\; f(B)$

(equal if $f$ is injective) , $f^\{-1\}(S\; \backslash setminus\; T)\; =\; f^\{-1\}(S)\; \backslash setminus\; f^\{-1\}(T)$ , - , $f\backslash left(A\; \backslash triangle\; B\backslash right)\; \backslash supseteq\; f(A)\; \backslash triangle\; f(B)$

(equal if $f$ is injective) , $f^\{-1\}\backslash left(S\; \backslash triangle\; T\backslash right)\; =\; f^\{-1\}(S)\; \backslash triangle\; f^\{-1\}(T)$ , - The results relating images and preimages to the ( Boolean) algebra of

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, the image of a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

is the set of all output values it may produce.
More generally, evaluating a given function $f$ at each element
Element may refer to:
Science
* Chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting only of atoms that all ...

of a given subset $A$ of its domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

produces a set, called the "image of $A$ under (or through) $f$". Similarly, the inverse image (or preimage) of a given subset $B$ of the codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of $f,$ is the set of all elements of the domain that map to the members of $B.$
Image and inverse image may also be defined for general binary relations
Binary may refer to:
Science and technology
Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...

, not just functions.
Definition

The word "image" is used in three related ways. In these definitions, $f\; :\; X\; \backslash to\; Y$ is afunction
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

from the set $X$ to the set $Y.$
Image of an element

If $x$ is a member of $X,$ then the image of $x$ under $f,$ denoted $f(x),$ is thevalue
Value or values may refer to:
* Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them
** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western s ...

of $f$ when applied to $x.$ $f(x)$ is alternatively known as the output of $f$ for argument $x.$
Given $y,$ the function $f$ is said to "" or "" if there exists some $x$ in the function's domain such that $f(x)\; =\; y.$
Similarly, given a set $S,$ $f$ is said to "" if there exists $x$ in the function's domain such that $f(x)\; \backslash in\; S.$
However, "" and "" means that $f(x)\; \backslash in\; S$ for point $x$ in $f$'s domain.
Image of a subset

Throughout, let $f\; :\; X\; \backslash to\; Y$ be a function. The under $f$ of a subset $A$ of $X$ is the set of all $f(a)$ for $a\backslash in\; A.$ It is denoted by $f;\; href="/html/ALL/s/.html"\; ;"title="">$ or by $f(A),$ when there is no risk of confusion. Usingset-builder notation
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

, this definition can be written as
$$f;\; href="/html/ALL/s/.html"\; ;"title="">$$
This induces a function $f;\; href="/html/ALL/s/,\backslash cdot\backslash ,.html"\; ;"title=",\backslash cdot\backslash ,">,\backslash cdot\backslash ,$ where $\backslash wp(S)$ denotes the power set
Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion.
In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ...

of a set $S;$ that is the set of all subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are u ...

s of $S.$ See below for more.
Image of a function

The ''image'' of a function is the image of its entiredomain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

, also known as the range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to ...

of the function. This usage should be avoided because the word "range" is also commonly used to mean the codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of $f.$
Generalization to binary relations

If $R$ is an arbitrarybinary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...

on $X\; \backslash times\; Y,$ then the set $\backslash $ is called the image, or the range, of $R.$ Dually, the set $\backslash $ is called the domain of $R.$
Inverse image

Let $f$ be a function from $X$ to $Y.$ The preimage or inverse image of a set $B\; \backslash subseteq\; Y$ under $f,$ denoted by $f^;\; href="/html/ALL/s/.html"\; ;"title="">$ is the subset of $X$ defined by $$f^;\; href="/html/ALL/s/B\_.html"\; ;"title="B\; ">B$$ Other notations include $f^(B)$ and $f^(B).$ The inverse image of asingleton set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, denoted by $f^;\; href="/html/ALL/s/.html"\; ;"title="">$fiber
Fiber or fibre (from la, fibra, links=no) is a #Natural fibers, natural or #Man-made fibers, man-made substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineer ...

or fiber over $y$ or the level set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...

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 $f(x)\; =\; x^2,$ the inverse image of $\backslash $ would be $\backslash .$ Again, if there is no risk of confusion, $f^;\; href="/html/ALL/s/.html"\; ;"title="">$inverse function
In mathematics, an inverse function (or anti-function) is a function (mathematics), function that "reverses" another function: if the function applied to an input gives a result of , then applying its inverse function to gives the result , i ...

, although it coincides with the usual one for bijections in that the inverse image of $B$ under $f$ is the image of $B$ under $f^.$
Notation for image and inverse image

The traditional notations used in the previous section may be confusing, because it does not distinguish the original function $f\; :\; X\; \backslash to\; Y$ from the image-of-sets function $f\; :\; \backslash mathcal(X)\; \backslash to\; \backslash mathcal(Y)$; likewise it does not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). An alternative is to give explicit names for the image and preimage as functions between power sets:Arrow notation

* $f^\backslash rightarrow\; :\; \backslash mathcal(X)\; \backslash to\; \backslash mathcal(Y)$ with $f^\backslash rightarrow(A)\; =\; \backslash $ * $f^\backslash leftarrow\; :\; \backslash mathcal(Y)\; \backslash to\; \backslash mathcal(X)$ with $f^\backslash leftarrow(B)\; =\; \backslash $Star notation

* $f\_\backslash star\; :\; \backslash mathcal(X)\; \backslash to\; \backslash mathcal(Y)$ instead of $f^\backslash rightarrow$ * $f^\backslash star\; :\; \backslash mathcal(Y)\; \backslash to\; \backslash mathcal(X)$ instead of $f^\backslash leftarrow$Other terminology

* An alternative notation for $f;\; href="/html/ALL/s/.html"\; ;"title="">$mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...

and set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...

is $f\backslash ,\text{'}\text{'}A.$M. Randall HolmesInhomogeneity of the urelements in the usual models of NFU

December 29, 2005, on: Semantic Scholar, p. 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of $f.$
Examples

# $f\; :\; \backslash \; \backslash to\; \backslash $ defined by $f(x)\; =\; \backslash left\backslash \{\backslash begin\{matrix\}\; a,\; \&\; \backslash mbox\{if\; \}x=1\; \backslash \backslash \; a,\; \&\; \backslash mbox\{if\; \}x=2\; \backslash \backslash \; c,\; \&\; \backslash mbox\{if\; \}x=3.\; \backslash end\{matrix\}\backslash right.$ The ''image'' of the set $\backslash \{\; 2,\; 3\; \backslash \}$ under $f$ is $f(\backslash \{\; 2,\; 3\; \backslash \})\; =\; \backslash \{\; a,\; c\; \backslash \}.$ The ''image'' of the function $f$ is $\backslash \{\; a,\; c\; \backslash \}.$ The ''preimage'' of $a$ is $f^\{-1\}(\backslash \{\; a\; \backslash \})\; =\; \backslash \{\; 1,\; 2\; \backslash \}.$ The ''preimage'' of $\backslash \{\; a,\; b\; \backslash \}$ is also $f^\{-1\}(\backslash \{\; 1,\; 2\; \backslash \})\; =\; \backslash \{\; 1,\; 2\; \backslash \}.$ The preimage of $\backslash \{\; b,\; d\; \backslash \},$ is theempty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical an ...

$\backslash \{\; \backslash ,\; \backslash \}\; =\; \backslash varnothing.$
# $f\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $f(x)\; =\; x^2.$ The ''image'' of $\backslash \{\; -2,\; 3\; \backslash \}$ under $f$ is $f^\{-1\}(\backslash \{\; -2,\; 3\; \backslash \})\; =\; \backslash \{\; 4,\; 9\; \backslash \},$ and the ''image'' of $f$ is $\backslash R^+$ (the set of all positive real numbers and zero). The ''preimage'' of $\backslash \{\; 4,\; 9\; \backslash \}$ under $f$ is $f^\{-1\}(\backslash \{\; 4,\; 9\; \backslash \})\; =\; \backslash \{\; -3,\; -2,\; 2,\; 3\; \backslash \}.$ The preimage of set $N\; =\; \backslash \{\; n\; \backslash in\; \backslash R\; :\; n\; <\; 0\; \backslash \}$ under $f$ is the empty set, because the negative numbers do not have square roots in the set of reals.
# $f\; :\; \backslash R^2\; \backslash to\; \backslash R$ defined by $f(x,\; y)\; =\; x^2\; +\; y^2.$ The ''fiber'' $f^\{-1\}(\backslash \{\; a\; \backslash \})$ are concentric circles
In geometry, two or more mathematical object, objects are said to be concentric, coaxal, or coaxial when they share the same center (geometry), center or Coordinate axis, axis. Circles, regular polygons and regular polyhedron, regular polyhedra, ...

about the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sla ...

, the origin itself, and the empty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical an ...

, depending on whether $a\; >\; 0,\; a\; =\; 0,\; \backslash text\{\; or\; \}\; a\; <\; 0,$ respectively. (if $a\; >\; 0,$ then the fiber $f^\{-1\}(\backslash \{\; a\; \backslash \})$ is the set of all $(x,\; y)\; \backslash in\; \backslash R^2$ satisfying the equation of the origin-concentric ring $x^2\; +\; y^2\; =\; a.$)
# If $M$ is a manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

and $\backslash pi\; :\; TM\; \backslash to\; M$ is the canonical projection from the tangent bundle
Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).
In differen ...

$TM$ to $M,$ then the ''fibers'' of $\backslash pi$ are the tangent spaces
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$T\_x(M)\; \backslash text\{\; for\; \}\; x\; \backslash in\; M.$ This is also an example of a fiber bundle
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

.
# A quotient group
A quotient group or factor group is a math
Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...

is a homomorphic image.
Properties

{, class=wikitable style="float:right;" , + ! Counter-examples based on thereal number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s $\backslash R,$$f\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $x\; \backslash mapsto\; x^2,$

showing that equality generally need

not hold for some laws: , - , , - , center, upright=1.7, $f\backslash left(f^\{-1\}\backslash left(B\_3\backslash right)\backslash right)\; \backslash subsetneq\; B\_3.$ , - ,

General

For every function $f\; :\; X\; \backslash to\; Y$ and all subsets $A\; \backslash subseteq\; X$ and $B\; \backslash subseteq\; Y,$ the following properties hold: {, class="wikitable" , - ! Image ! Preimage , - , $f(X)\; \backslash subseteq\; Y$ , $f^\{-1\}(Y)\; =\; X$ , - , $f\backslash left(f^\{-1\}(Y)\backslash right)\; =\; f(X)$ , $f^\{-1\}(f(X))\; =\; X$ , - , $f\backslash left(f^\{-1\}(B)\backslash right)\; \backslash subseteq\; B$(equal if $B\; \backslash subseteq\; f(X);$ for instance, if $f$ is surjective)See See , $f^\{-1\}(f(A))\; \backslash supseteq\; A$

(equal if $f$ is injective) , - , $f(f^\{-1\}(B))\; =\; B\; \backslash cap\; f(X)$ , $\backslash left(f\; \backslash vert\_A\backslash right)^\{-1\}(B)\; =\; A\; \backslash cap\; f^\{-1\}(B)$ , - , $f\backslash left(f^\{-1\}(f(A))\backslash right)\; =\; f(A)$ , $f^\{-1\}\backslash left(f\backslash left(f^\{-1\}(B)\backslash right)\backslash right)\; =\; f^\{-1\}(B)$ , - , $f(A)\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; A\; =\; \backslash varnothing$ , $f^\{-1\}(B)\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; B\; \backslash subseteq\; Y\; \backslash setminus\; f(X)$ , - , $f(A)\; \backslash supseteq\; B\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\; \backslash text\{\; there\; exists\; \}\; C\; \backslash subseteq\; A\; \backslash text\{\; such\; that\; \}\; f(C)\; =\; B$ , $f^\{-1\}(B)\; \backslash supseteq\; A\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f(A)\; \backslash subseteq\; B$ , - , $f(A)\; \backslash supseteq\; f(X\; \backslash setminus\; A)\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f(A)\; =\; f(X)$ , $f^\{-1\}(B)\; \backslash supseteq\; f^\{-1\}(Y\; \backslash setminus\; B)\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f^\{-1\}(B)\; =\; X$ , - , $f(X\; \backslash setminus\; A)\; \backslash supseteq\; f(X)\; \backslash setminus\; f(A)$ , $f^\{-1\}(Y\; \backslash setminus\; B)\; =\; X\; \backslash setminus\; f^\{-1\}(B)$ , - , $f\backslash left(A\; \backslash cup\; f^\{-1\}(B)\backslash right)\; \backslash subseteq\; f(A)\; \backslash cup\; B$See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. , $f^\{-1\}(f(A)\; \backslash cup\; B)\; \backslash supseteq\; A\; \backslash cup\; f^\{-1\}(B)$ , - , $f\backslash left(A\; \backslash cap\; f^\{-1\}(B)\backslash right)\; =\; f(A)\; \backslash cap\; B$ , $f^\{-1\}(f(A)\; \backslash cap\; B)\; \backslash supseteq\; A\; \backslash cap\; f^\{-1\}(B)$ Also: * $f(A)\; \backslash cap\; B\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; A\; \backslash cap\; f^\{-1\}(B)\; =\; \backslash varnothing$

Multiple functions

For functions $f\; :\; X\; \backslash to\; Y$ and $g\; :\; Y\; \backslash to\; Z$ with subsets $A\; \backslash subseteq\; X$ and $C\; \backslash subseteq\; Z,$ the following properties hold: * $(g\; \backslash circ\; f)(A)\; =\; g(f(A))$ * $(g\; \backslash circ\; f)^\{-1\}(C)\; =\; f^\{-1\}(g^\{-1\}(C))$Multiple subsets of domain or codomain

For function $f\; :\; X\; \backslash to\; Y$ and subsets $A,\; B\; \backslash subseteq\; X$ and $S,\; T\; \backslash subseteq\; Y,$ the following properties hold: {, class="wikitable" , - ! Image ! Preimage , - , $A\; \backslash subseteq\; B\; \backslash ,\backslash text\{\; implies\; \}\backslash ,\; f(A)\; \backslash subseteq\; f(B)$ , $S\; \backslash subseteq\; T\; \backslash ,\backslash text\{\; implies\; \}\backslash ,\; f^\{-1\}(S)\; \backslash subseteq\; f^\{-1\}(T)$ , - , $f(A\; \backslash cup\; B)\; =\; f(A)\; \backslash cup\; f(B)$ , $f^\{-1\}(S\; \backslash cup\; T)\; =\; f^\{-1\}(S)\; \backslash cup\; f^\{-1\}(T)$ , - , $f(A\; \backslash cap\; B)\; \backslash subseteq\; f(A)\; \backslash cap\; f(B)$(equal if $f$ is injectiveSee ) , $f^\{-1\}(S\; \backslash cap\; T)\; =\; f^\{-1\}(S)\; \backslash cap\; f^\{-1\}(T)$ , - , $f(A\; \backslash setminus\; B)\; \backslash supseteq\; f(A)\; \backslash setminus\; f(B)$

(equal if $f$ is injective) , $f^\{-1\}(S\; \backslash setminus\; T)\; =\; f^\{-1\}(S)\; \backslash setminus\; f^\{-1\}(T)$ , - , $f\backslash left(A\; \backslash triangle\; B\backslash right)\; \backslash supseteq\; f(A)\; \backslash triangle\; f(B)$

(equal if $f$ is injective) , $f^\{-1\}\backslash left(S\; \backslash triangle\; T\backslash right)\; =\; f^\{-1\}(S)\; \backslash triangle\; f^\{-1\}(T)$ , - The results relating images and preimages to the ( Boolean) algebra of

intersection
The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points.
In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...

and union work for any collection of subsets, not just for pairs of subsets:
* $f\backslash left(\backslash bigcup\_\{s\backslash in\; S\}A\_s\backslash right)\; =\; \backslash bigcup\_\{s\backslash in\; S\}\; f\backslash left(A\_s\backslash right)$
* $f\backslash left(\backslash bigcap\_\{s\backslash in\; S\}A\_s\backslash right)\; \backslash subseteq\; \backslash bigcap\_\{s\backslash in\; S\}\; f\backslash left(A\_s\backslash right)$
* $f^\{-1\}\backslash left(\backslash bigcup\_\{s\backslash in\; S\}B\_s\backslash right)\; =\; \backslash bigcup\_\{s\backslash in\; S\}\; f^\{-1\}\backslash left(B\_s\backslash right)$
* $f^\{-1\}\backslash left(\backslash bigcap\_\{s\backslash in\; S\}B\_s\backslash right)\; =\; \backslash bigcap\_\{s\backslash in\; S\}\; f^\{-1\}\backslash left(B\_s\backslash right)$
(Here, $S$ can be infinite, even uncountably infinite
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.)
With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a semilatticeIn mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilattic ...

homomorphism (that is, it does not always preserve intersections).
See also

* * * *Notes

References

* * . * * * * {{PlanetMath attribution, id=3276, title=FibreBasic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...

Isomorphism theorems