Somehow the notion of a function (between sets) play an important role and have contributed to forming concepts in mathematics. Sometimes I wonder how mathematics would have evolved if we used relations instead of functions.

Often you see proofs that under certain assumption something is well defined and they you define a function mapping something to that element. If the assumptions do not hold, then you can't continue to work. In the setting of relations, we would then map the source element to all of the choices.