Difference between revisions of "Functions:Forward Image"
| Line 8: | Line 8: | ||
The word "image" is used in three related ways. In these definitions, <math>f:X\to\ Y</math> is a function from the set <math>X</math> to the set <math>Y</math>. | The word "image" is used in three related ways. In these definitions, <math>f:X\to\ Y</math> is a function from the set <math>X</math> to the set <math>Y</math>. | ||
| − | Image of an element | + | ===Image of an element=== |
| − | If | + | If <math>x</math> is a member of <math>X</math>, then the image of <math>x</math> under <math>f</math>, denoted <math>f(x)</math>, is the value of <math>f</math> when applied to <math>x</math>. <math>f(x)</math> is alternatively known as the output of <math>f</math> for argument <math>x</math>. |
| − | Given | + | Given <math>y</math>, the function <math>f</math> is said to "take the value <math>y</math>" or "take <math>y</math> as a value" if there exists some <math>x</math> in the function's domain such that <math>f(x)=y</math>. Similarly, given a set <math>S</math>, <math>f</math> is said to "take a value in <math>S</math>" if there exists some <math>x</math> in the function's domain such that <math>f(x)\in S</math>. However, "<math>f</math> takes all values in <math>S</math>" and "<math>f</math> is valued in <math>S</math>" means that <math>f(x)\in S</math> for every point <math>x</math> in <math>f</math>'s domain. |
| − | Image of a subset | + | ===Image of a subset=== |
The image of a subset {\displaystyle A\subseteq X}A\subseteq X under {\displaystyle f,}f, denoted {\displaystyle f[A],}{\displaystyle f[A],} is the subset of {\displaystyle Y}Y which can be defined using set-builder notation as follows:[1][2] | The image of a subset {\displaystyle A\subseteq X}A\subseteq X under {\displaystyle f,}f, denoted {\displaystyle f[A],}{\displaystyle f[A],} is the subset of {\displaystyle Y}Y which can be defined using set-builder notation as follows:[1][2] | ||
| Line 19: | Line 19: | ||
When there is no risk of confusion, {\displaystyle f[A]}{\displaystyle f[A]} is simply written as {\displaystyle f(A).}{\displaystyle f(A).} This convention is a common one; the intended meaning must be inferred from the context. This makes {\displaystyle f[\,\cdot \,]}{\displaystyle f[\,\cdot \,]} a function whose domain is the power set of {\displaystyle X}X (the set of all subsets of {\displaystyle X}X), and whose codomain is the power set of {\displaystyle Y.}Y. See § Notation below for more. | When there is no risk of confusion, {\displaystyle f[A]}{\displaystyle f[A]} is simply written as {\displaystyle f(A).}{\displaystyle f(A).} This convention is a common one; the intended meaning must be inferred from the context. This makes {\displaystyle f[\,\cdot \,]}{\displaystyle f[\,\cdot \,]} a function whose domain is the power set of {\displaystyle X}X (the set of all subsets of {\displaystyle X}X), and whose codomain is the power set of {\displaystyle Y.}Y. See § Notation below for more. | ||
| − | Image of a function | + | ===Image of a function=== |
The image of a function is the image of its entire domain, also known as the range of the function.[3] This usage should be avoided because the word "range" is also commonly used to mean the codomain of {\displaystyle f.}f. | The image of a function is the image of its entire domain, also known as the range of the function.[3] This usage should be avoided because the word "range" is also commonly used to mean the codomain of {\displaystyle f.}f. | ||
| − | Generalization to binary relations | + | ===Generalization to binary relations=== |
If {\displaystyle R}R is an arbitrary binary relation on {\displaystyle X\times Y,}{\displaystyle X\times Y,} then the set {\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}}{\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}} is called the image, or the range, of {\displaystyle R.}R. Dually, the set {\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}}{\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}} is called the domain of {\displaystyle R.}R. | If {\displaystyle R}R is an arbitrary binary relation on {\displaystyle X\times Y,}{\displaystyle X\times Y,} then the set {\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}}{\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}} is called the image, or the range, of {\displaystyle R.}R. Dually, the set {\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}}{\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}} is called the domain of {\displaystyle R.}R. | ||
Revision as of 09:27, 12 October 2021
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function at each element of a given subset of its domain produces a set, called the "image of under (or through) ". Similarly, the inverse image (or preimage) of a given subset of the codomain of , is the set of all elements of the domain that map to the members of .
Image and inverse image may also be defined for general binary relations, not just functions.
Contents
Definition
The word "image" is used in three related ways. In these definitions, is a function from the set to the set .
Image of an element
If is a member of , then the image of under , denoted , is the value of when applied to . is alternatively known as the output of for argument .
Given , the function is said to "take the value " or "take as a value" if there exists some in the function's domain such that . Similarly, given a set , is said to "take a value in " if there exists some in the function's domain such that . However, " takes all values in " and " is valued in " means that for every point in 's domain.
Image of a subset
The image of a subset {\displaystyle A\subseteq X}A\subseteq X under {\displaystyle f,}f, denoted {\displaystyle f[A],}{\displaystyle f[A],} is the subset of {\displaystyle Y}Y which can be defined using set-builder notation as follows:[1][2]
{\displaystyle f[A]=\{f(x):x\in A\}}{\displaystyle f[A]=\{f(x):x\in A\}} When there is no risk of confusion, {\displaystyle f[A]}{\displaystyle f[A]} is simply written as {\displaystyle f(A).}{\displaystyle f(A).} This convention is a common one; the intended meaning must be inferred from the context. This makes {\displaystyle f[\,\cdot \,]}{\displaystyle f[\,\cdot \,]} a function whose domain is the power set of {\displaystyle X}X (the set of all subsets of {\displaystyle X}X), and whose codomain is the power set of {\displaystyle Y.}Y. See § Notation below for more.
Image of a function
The image of a function is the image of its entire domain, also known as the range of the function.[3] This usage should be avoided because the word "range" is also commonly used to mean the codomain of {\displaystyle f.}f.
Generalization to binary relations
If {\displaystyle R}R is an arbitrary binary relation on {\displaystyle X\times Y,}{\displaystyle X\times Y,} then the set {\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}}{\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}} is called the image, or the range, of {\displaystyle R.}R. Dually, the set {\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}}{\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}} is called the domain of {\displaystyle R.}R.
