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If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows:
for all x in X.
This turns the set of all such functions into a ring. The binary operations in that ring have as domain ordered pairs of functions, and as codomain functions. This is an example of climbing up in abstraction, to functions of more complex types.
By taking some other algebraic structure A in the place of R, we can turn the set of all functions from X to A into an algebraic structure of the same type in an analogous way.
The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. This argument shows that there are functions from integers to integers that are not computable. For examples of noncomputable functions, see the articles on the halting problem and Rice's theorem.
In the context of category theory, a function no longer represents a rule for taking an input to an output, but instead represents a relationship between its domain and its codomain. Since these functions are no longer functions in the usual sense, they are usually referred to as morphisms. A morphism is then an ordered triple (X, Y, f), where f is a "function" with domain X and codomain Y. Since X and Y do not necessarily correspond to a set of objects, however, morphisms do not always behave like functions, and, for example, enlarging the codomain (which does nothing to a function) gives a different morphism which you cannot identify with the original one.
Ordinary functions are sometimes referred to as morphisms when they are morphisms in a concrete category.
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