Is restriction of a function necessarily limited to a subset of the domain?

Question

A pretty basic question but I cannot find it either here or on the net.

The usual definition of restriction of a function $f : X \to Y$, to a set $X'$ (denoted by $f{\restriction_{X'}}$) assumes that $X' \subset X$, i.e. the restricted domain is a subset of the original domain. See for example the Wikipedia definition of restriction.

However, in certain contexts it may be simpler and very natural to extend this concept to any set $X'$, simply by considering $X \cap X'$; thus writing $f{\restriction_{X'}}$ directly instead of $f{\restriction_{X' \cap X}}$, because:

• the intent is clear even if $X'$ is not a subset of the domain;
• it saves some characters;
• if the domain of the function has no established name, than this operation requires even more characters, (something like $f{\restriction_{X' \cap {\operatorname {dom} f} }}$);
• if the function is constructed as an expression (like $g\circ h$), then this becomes even more complex and requires unnecessary repetition $(g \circ h){\restriction_{X' \cap \operatorname {dom}(g \circ h) }}$ .

Naturally, if such expressions become abundant, it is possible to add a short remark on the extended usage. However, I am still curious whether

• such extension would be considered (even a minor) abuse of notation without any remark?
• there is some deeper reason why the "usual" definition assumes that the restriction is limited to subsets of the domain? This constraint seems completely unnecessary.

Note:

Despite using the "usual" definition, even the Wikipedia article hints at the possibility of the "extended" one:

Informally, the restriction of $f$ to $A$ is the same function as $f$, but is only defined on $\displaystyle A\cap \operatorname {dom} f$.

because if we had really assumed $A \subset \operatorname {dom} f$ then it would have been enough to write "... but is only defined on $A$".

Answer 1

It doesn't really matter. Let $f:X\to Y$ be a function and $X'$ be a set. Define the restriction of $f$ to $X'$ as $f\restriction_{X'}=\{(x,y)\in f:x\in X'\}$.

You'll find that it agrees with the usual definition whenever $X'\subseteq X$. Else, $X'\nsubseteq X$ and you'll find that $f\restriction_{X'}=f\restriction_{X'\cap X}$. FWIW I prefer my definition as the usual one is unnecessarily restrictive, and as you said, shortens notation.