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| \textbf{Definition}. Let $f:A\to B$ and $g:C\to D$ be partial functions. $g$ is said to be a \emph{subfunction} of $f$ if $$g\subseteq f \cap (C\times D).$$ |
\textbf{Definition}. Let $f:A\to B$ and $g:C\to D$ be partial functions. $g$ is said to be a \emph{subfunction} of $f$ if $$g\subseteq f \cap (C\times D).$$ |
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| 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 $g(x)=f(x)$. |
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 $g(x)=f(x)$. |
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| If we set $C'=A\cap C$ and $D'=B\cap D$, then $g\subseteq f\cap (C'\times D')$, 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. |
If we set $C'=A\cap C$ and $D'=B\cap D$, then $g\subseteq f\cap (C'\times D')$, 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. |
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| 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 $D$. |
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 $D$. |
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| 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 |
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 $(-\infty,-1)\cup (1,\infty)$. |
$$g(x)=\displaystyle{\frac{x^2-1}{\sqrt{x^2-1}}}$$ is a subfunction of $f$. The domain of definition of $g$ is $(-\infty,-1)\cup (1,\infty)$. |
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| Two immediate properties of a subfunction $g:C\to D$ of $f:A\to B$ are |
Two immediate properties of a subfunction $g:C\to D$ of $f:A\to B$ are |
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\begin{itemize} |
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| the range of $g$ is a subset of the range of $f$: $$g(C)\subseteq f(C),$$ |
the range of $g$ is a subset of the range of $f$: $$g(C)\subseteq f(C),$$ |
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\item |
| the domain of definition of $g$ is a subset of the domain of definition of $f$: $$g^{-1}(D)\subseteq f^{-1}(D).$$ |
the domain of definition of $g$ is a subset of the domain of definition of $f$: $$g^{-1}(D)\subseteq f^{-1}(D).$$ |
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\end{itemize} |
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| \textbf{Definition}. A subfunction $g:C\to D$ of $f:A\to B$ is called a \emph{restriction of $f$ relative to $D$}, if $g(C)=f(C)\cap D$, and a \emph{restriction of $f$} if $g(C)=f(C)$. |
\textbf{Definition}. A subfunction $g:C\to D$ of $f:A\to B$ is called a \emph{restriction of $f$ relative to $D$}, if $g(C)=f(C)\cap D$, and a \emph{restriction of $f$} if $g(C)=f(C)$. |
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| 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. |
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. |
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| More to come... |
More to come... |
