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Homesubfunction
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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 nonnegative 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).
Mathematics Subject Classification
08A55 no label found03E20 no label found Forums
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