# 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: , 2nd Edition, Springer, New York (1978).
Title subfunction Subfunction 2013-03-22 18:41:54 2013-03-22 18:41:54 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 08A55 msc 03E20 restriction