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Function composition is defined on Wolfram to be:

'The nesting of two or more functions to form a single new function is known as composition. The composition of two functions $f$ and $g$ is denoted $f \circ g$, where $f$ is a function whose domain includes the range of $g$.'

I would like to ask why isn't composition defined to be something like the following instead:

Function composition, denoted $\circ$, is a partial binary operation on the set of all functions such that $(f \circ g)(x)=f(g(x))$ for all functions $f$, $g$ where $\mathscr{R}(g)\subseteq\mathscr{D}(f)$.

Is it because the set of all functions isn't well defined?

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  • $\begingroup$ It is fine to view composition as a binary operation on a certain set of functions. (In practice, we often need consider the set of functions from a set $A$ into a set $A$, for example; then composition is a binary operation on this set.) Why Wolfram chose not to take this point of view formally/explicitly is not really a mathematical question. $\endgroup$ Commented Jul 4, 2017 at 6:23

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"All functions" is a proper class and can't be represented as a set. However defining composition as a partial binary operation on a class of functions is a legitimate definition.

Indeed if you define composition as a closed associative partial binary operation on a class of morphisms (a function is a type of morphism) you more or less have the category theory definition of composition.

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