I would like to argue that the difference between the two approaches is essentially cosmetic, and which definition of "function" you choose matters very rarely. In the first approach, it does not make sense to ask whether a function $f$ is surjective; after all, if $f$ is a function $X\to Y$, then given any set $Z$ such that $Y\subseteq Z$, it is also the case that $f$ is a function $X\to Z$. This is only a minor inconvenience, though: you can still talk about a function $f:X\to Y$ being surjective – you just have to say that $f$ surjects onto the set $Y$, rather than simply saying $f$ is surjective. Meanwhile, the first approach has the minor convenience that if we identify functions with their graphs (as is standard in axiomatic set theory), then we can talk about unions of functions – this is rather common in the context of axiomatic set theory, but less so elsewhere in mathematics.
I would like to argue that the difference between the two approaches is essentially cosmetic, and which definition of "function" you choose matters very rarely. In the first approach, it does not make sense to ask whether a function $f$ is surjective; after all, if $f$ is a function $X\to Y$, then given any set $Z$ such that $Y\subseteq Z$, it is also the case that $f$ is a function $X\to Z$. This is only a minor inconvenience, though: you can still talk about a function $f:X\to Y$ being surjective – you just have to say that $f$ surjects onto the set $Y$, rather than simply saying $f$ is surjective. Meanwhile, the first approach has the minor convenience that if we identify functions with their graphs (as is standard in axiomatic set theory), then we can talk about unions of functions – this is rather common in the context of axiomatic set theory, but less so elsewhere in mathematics.
I would like to argue that the difference between the two approaches is essentially cosmetic, and which definition of "function" you choose matters very rarely. In the first approach, it does not make sense to ask whether a function $f$ is surjective; after all, if $f$ is a function $X\to Y$, then given any set $Z$ such that $Y\subseteq Z$, it is also the case that $f$ is a function $X\to Z$. This is only a minor inconvenience, though: you can still say that $f$ surjects onto the set $Y$, rather than simply saying $f$ is surjective. Meanwhile, the first approach has the minor convenience that if we identify functions with their graphs (as is standard in axiomatic set theory), then we can talk about unions of functions – this is rather common in the context of axiomatic set theory, but less so elsewhere in mathematics.
I would like to argue that the difference between the two approaches is essentially cosmetic, and which definition of "function" you choose matters very rarely. In the first approach, it does not make sense to ask whether a function $f$ is surjective; after all, if $f$ is a function $X\to Y$, then given any set $Z$ such that $Y\subseteq Z$, it is also the case that $f$ is a function $X\to Z$. This is only a minor inconvenience, though: you can still talk about a function $f:X\to Y$ being surjective – you just have to say that $f$ surjects onto the set $Y$, rather than simply saying $f$ is surjective. Meanwhile, the first approach has the minor convenience that if we identify functions with their graphs (as is standard in axiomatic set theory), then we can talk about unions of functions – this is rather common in the context of axiomatic set theory, but less so elsewhere in mathematics.