Update Mathematics of Functions (New Math)
Functions by definition are relations, which are subsets of a Cartesian product, but our current definition of a function is missing the fact that functions are sets of 0 or more component functions (operations) along with the relations between the domain and those components, the relations between the range and those components, and the relations of those components to one another. The set of component functions can have properties such as order to name one. Let's discuss this topic. I look forward to your thoughts on this.
Wolfram mathworld states:
function A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from A to B is an object f such that every a in A is uniquely associated with an object f(a) in B. A function is therefore a many-to-one (or sometimes one-to-one) relation.
relation A relation is any subset of a Cartesian product. For instance, a subset of A×B, called a "binary relation from A to B," is a collection of ordered pairs (a,b) with first components from A and second components from B, and, in particular, a subset of A×A is called a "relation on A." For a binary relation R, one often writes aRb to mean that (a,b) is in R×R.