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Suppose we have a function $f:X \rightarrow Y$. Now, consider the function $g:X'\rightarrow Y$ where $X'\subset X$.

I'd like to say the $g$ is a "subset" of $f$ ; is there a correct term for describing $g$ w.r.t $f$?

NB: Also, $g(x) = f(x)$ for $x \in X'$

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2 Answers 2

up vote 5 down vote accepted

Provided $g(x)=f(x)$ for all $x \in X'$, we say that $g$ is the restriction of $f$ to $X'$, and that $f$ is an extension of $g$ to $X$.

If $g$ is the restriction of $f$ to $X'$ we can write $g=f|_{X'}$.

Restrictions are unique (hence 'the' restriction), extensions are not (hence 'an' extension).

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When I say "$g$ is a restriction of $f$ to $X'$", is it understood that $X' \subseteq X$? –  Jacob Sep 14 '12 at 18:10
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@P23: Yes, the word 'restriction' is only used when $X' \subseteq X$, but it's good practice to mention the fact that $X' \subseteq X$ too. –  Clive Newstead Sep 14 '12 at 18:13
    
Thanks! Also (just checking), $g(x)=f(x)$ for $x \in X'$? –  Jacob Sep 14 '12 at 18:13
    
@P23: Yes, otherwise $g$ and $f$ might bear no relation to each other! There are functions $g : X' \to Y$ for $X' \subseteq X$ which are not restrictions of $X$. –  Clive Newstead Sep 14 '12 at 18:14
    
Just checking (just in case the term was for functions where the domains were related and codomains were the same). Thanks, again! –  Jacob Sep 14 '12 at 18:15

What you are referring to is the restriction of $f$ to $X'$ — assuming that you define $g(x) =f(x)$ your all $x \in X'$, that is.

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