Empty Set Problem

[Ender]

No, a function is defined as a set. A function f : A -> B is defined as a subset of A x B (where the set A x B is the direct product of A and B) s.t. for each a in A there exists a unique b in B with (a, b) in f.

EDIT: Pretty much everything in math is derived from sets (hence Set Theory), although Category Theory messes things up a bit.

The set described by a function does describe the function, but it does not identify it! In set theory, it is not a goal to return to the original function, although it's almost always trivial to do so, because of the simple fact that distinct functions with equivalent input-output are distinct only in their definition, and not in their behavior.

Set theory is nothing more than a mental aid for mathematics; it isn't required for deriving anything. Because it's based simpler principles, it's possible to bypass it. Doing so would be equivalent to reducing an integral to a summation, or a derivative to a limit.

No, you're wrong. f is defined as a set of ordered pairs, viz. {(a, b), (c, d),...}. The inverse function is a set of the same form. Properties of functions like surjectivity, injectivity, and bijectivity are easy to define with sets. The natural numbers are constructed with sets, and the reals too, although there's more to it, like Dedekind cuts.

[Camel]

Okay, so I actually did some research in to this to settle the topic.

The definition of a function is the rule that describes the relationship between the elements of its range and the elements of its domain. The tricky part is that there's controversy over how the word "rule" should be defined. One interpretation of the word rule is what you said: the binary relation its self. The other school of thought says that the binary relation must come from the function, otherwise it's just an arbitrary set with no definition.

Technically, we're both correct under different definitions of the word "rule," although this is totally irrelevant in the context of the thread, so back to the topic:

The asymptote is the representation of the thing that is missing!

Also, it doesn't make much sense to describe a subset of an empty set, lol.