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, $f:ℝ\to ℝ$ defined by

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

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

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

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

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 $n{-}_{A}n=0\in A$.