subfunction

Definition. Let $f:A\to B$ and $g:C\to D$ be partial functions. $g$ is said to be a subfunction of $f$ if

$g\subseteq f\cap(C\times D).$

In other words, $g$ is a subfunction of $f$ iff whenever $x\in C$ such that $g(x)$ is defined, then $x\in A$ , $f(x)$ is defined, and $g(x)=f(x)$ .

If we set $C^{\prime}=A\cap C$ and $D^{\prime}=B\cap D$ , then $g\subseteq f\cap(C^{\prime}\times D^{\prime})$ , so there is no harm in assuming that $C$ and $D$ are subsets of $A$ and $B$ respectively, which we will do for the rest of the discussion.

In practice, whenever $g$ is a subfunction of $f$ , we often assume that $g$ and $f$ have the same domain and codomain. Otherwise, we would specify that $g$ is a subfunction of $f:A\to B$ with domain $C$ and codomain $D$ .

For example, $f:\mathbb{R}\to\mathbb{R}$ defined by

$f(x)=\sqrt{x^{2}-1}$

is a partial function, whose domain of definition is $(-\infty,-1]\cup[1,\infty)$ , and the partial function $g:\mathbb{R}\to\mathbb{R}$ given by

$g(x)=\displaystyle{\frac{x^{2}-1}{\sqrt{x^{2}-1}}}$

is a subfunction of $f$ . The domain of definition of $g$ is $(-\infty,-1)\cup(1,\infty)$ .

Two immediate properties of a subfunction $g:C\to D$ of $f:A\to B$ are

•

the range of $g$ is a subset of the range of $f$ :

$g(C)\subseteq f(C),$

•

the domain of definition of $g$ is a subset of the domain of definition of $f$ :

$g^{-1}(D)\subseteq f^{-1}(D).$

Definition. A subfunction $g:C\to D$ of $f:A\to B$ is called a restriction of $f$ relative to $D$ , if $g(C)=f(C)\cap D$ , and a restriction of $f$ if $g(C)=f(C)$ .

Every partial function $g:C\to D$ corresponds to a unique restriction $g^{\prime}:C\to g(C)$ of $g$ .

A restriction $g:C\to D$ of $f:A\to B$ is certainly a restriction of $f$ relative to $D$ , since $f(C)\cap D=g(C)\cap D=g(C)$ , but not conversely. For example, let $A$ be the set of all non-negative integers and $-_{A}:A^{2}\to A$ the ordinary subtraction. $-_{A}$ is easily seen to be a partial function. Let $B$ be the set of all positive integers. Then $-_{B}:B^{2}\to B$ is a restriction of $-_{A}:A^{2}\to A$ , relative to $B$ . However, $-_{B}$ is not a restriction of $-_{A}$ , for $n-_{B}n$ is not defined, while $n-_{A}n=0\in A$ .

References

1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).

Title	subfunction
Canonical name	Subfunction
Date of creation	2013-03-22 18:41:54
Last modified on	2013-03-22 18:41:54
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Definition
Classification	msc 08A55
Classification	msc 03E20
Defines	restriction