In general, relations not always create an unambiguous relation between set elements, i.e. the element of one set is related to more than one element of another set. Also, in general, relations not always relate all elements of a set to another set. If a relation fulfills these "nice" a properties - the right-uniqueness and left-totalilty - it is called a function or a map. Functions are an essential mathematical tool to study the dependencies between set elements and have therefore various applications in and outside mathematics we will learn examples of later.
In many cases, however, the nice-to-have property of being left-total is not fulfilled, but we still want to study the dependencies for at least a subset of the domain of our interest. Functions which are only right-unique, but are not left-total, are called partial functions, and we will learn some examples of this kind of maps as well.
But before we proceed with some examples and applications, let us more formally define functions, partial functions, and all related terms playing an important role when we deal with them: