- function, logical
- In logic and mathematics a function, also known as a map or mapping, is a relation that associates members of one class X with some unique member
*y*of another class Y. The association is written as*y*= f(*x*). The class X is called the domain of the function, and Y its range. Thus ‘the father of*x*’ is a function whose domain includes all people, and whose range is the class of male parents. But the relation ‘son of*x*’ is not a function, because a person can have more than one son. ‘Sine*x*’ is a function from angles onto real numbers; the length of the perimeter of a circle, π*x*, is a function of its diameter*x*; and so on. Functions may take sequences <*x*_{1}…*x*_{n}> as their arguments, in which case they may be thought of as associating a unique member of Y with any ordered n-tuple as argument. Given the equation*y*= f(*x*_{1}…*x*_{n}),*x*_{1}…*x*_{n}are called the independent variables, or arguments of the function, and*y*the dependent variable or value. Functions may be many-one, meaning that different members of X may take the same member of Y as their value, or one-one, when to each member of X there corresponds a distinct member of Y. A function with domain X and range Y is also called a mapping from X to Y, written f X → Y. If the function is such that(i) if*x*,*y*∈ X and f(*x*) = f(*y*) then*x*=*y*, then the function is an injection from X to Y. If also(ii) if*y*∈ Y, then (∃*x*)(*x*∈ X &*y*= f(*x*)) then the function is a bijection of X onto Y. A bijection is also known as a one-one correspondence. A bijection is both an injection and a surjection where a surjection is any function whose domain is X and whose range is the whole of Y. Since functions are relations a function may be defined as a set of ordered pairs <*x*,*y*> where*x*is a member of X and*y*of Y.One of Frege's logical insights was that a concept is analagous to a function, and a predicate analagous to the expression for a function (a functor). Just as ‘the square root of*x*’ takes us from one number to another, so ‘*x*is a philosopher’ refers to a function that takes us from persons to truth-values: true for values of*x*who are philosophers, and false otherwise.

*Philosophy dictionary.
Academic.
2011.*

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