Pictures in bulk
Formulas are nothing more than mappings. This can take many forms.
The word archetype sounds like something that might be considered in art history. But you don't, because a prototype looks like this:
Suppose you have a function that maps elements of set A to set B. A is the definition set, i.e. all the objects to which the function can be applied. B is the target set, i.e. all possible objects that can in principle be the result of the arithmetic operation specified by the function.
This morning, for example, I was looking for new running routes in my area. In order to get an overview, I first estimated on a map how far away certain places are in a direct line from my apartment. So I took points in the 2D plane and gave them a distance. In more mathematical terms, the definition set here corresponds to all pairs of numbers (x, y) in the Euclidean plane and the target set corresponds to all real numbers. Since the distance is given by the function f(x, y)=(x2+y2)½ is calculated and can never be negative, but not all possible real numbers are actually accepted as function values. So the image set of my function is smaller than the target set.
This brings us back to the concept of the archetype. As for my running distances, I don't want to run every possible distance; most of them would be way too long. If I select all points on the map that are exactly ten kilometers away from my home, I get a subset of the definition set. All elements in this subset are mapped to element 10 kilometers in the target set.
What goes with what?
All these possible targets form the archetype of element 10. Or in general: Given a function f that maps certain elements of set A onto elements of set B, the set given by the above formula is the archetype of M under f. So the set of all elements x from the definition set A for which the function f(x) is part of M.
The archetype is an important concept when trying to understand the properties of mathematical functions. For example, if each element of the target set has at most one preimage in the domain, then the function is called injective. An example of this would be the function f(x)=x2, as long as it refers to the natural numbers: no two different natural numbers multiplied by themselves can give the same result. For the integers, however, f(x)=x2 is no longer injective, since f(-4)=f(4)=16; in this case element 16 of the target set has two preimages (or the set of preimages comprises two elements).
Quasi the counterpart to injective functions are surjective functions. Here every element in the target set has at least one preimage. Each element of the target set is thus assumed at least once as a function value by the function. The function f(x)=x2 is not surjective if both the definition and the target set are given by the real numbers. For negative numbers then have no archetype; no real number squared can produce a negative result.
If a function is both injective and surjective, then it is called bijective. Exactly one element of the target set is assigned to each element of the definition set. No value of the target set is accepted more than once, and each element of the target set is accepted. For example, the square function would be bijective if both the definition and target sets are nonnegative real numbers.
The terms injective, surjective and bijective were introduced in the 1950s to finally get order in the terminology used to describe functions. For those who still find it all confusing, take comfort in the fact that it used to be a lot more chaotic.
