# Functions

### Functions, injections, surjections, bijections, one-to-one, onto, partial functions, total functions, compositions of functions, compositions of relations, isomorphisms

What is a function, precisely? Well, the essential property of functions is that for every suitable choice of input object, the function determines a corresponding output object, the value of the function on that input. So a function would seem to be a relation of a certain kind, relating the input objects to their output values. Indeed, we may simply define a function to be a special kind of binary relation, a *functional* relation, which means:

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