partial function - meaning and definition. What is partial function
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What (who) is partial function - definition

BINARY RELATION WHOSE ACTUAL DOMAIN MAY BE SMALLER THAN ITS APPARENT DOMAIN
Total function; Domain of definition; Partial mapping; Limited function; Limited function (mathematics); Partial functions; Partial Function; Total functions; ⇸; Natural domain; Domain of a partial function; Total Function; Partially-defined map; Partially defined map; Partial and total functions; Partial map

partial function         
A function which is not defined for all arguments of its input type. E.g. f(x) = 1/x if x /= 0. The opposite of a total function. In {denotational semantics}, a partial function f : D -> C may be represented as a total function ft : D' -> lift(C) where D' is a superset of D and ft x = f x if x in D ft x = bottom otherwise where lift(C) = C U bottom. Bottom (LaTeX perp) denotes "undefined". (1995-02-03)
Partial function         
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of .
total function         
<mathematics> A function which is defined for all arguments of the appropriate type. The opposite is a {partial function}. (1997-01-10)

Wikipedia

Partial function

In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function.

More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to exactly one element of the second set.

A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total.

When arrow notation is used for functions, a partial function f {\displaystyle f} from X {\displaystyle X} to Y {\displaystyle Y} is sometimes written as f : X Y , {\displaystyle f:X\rightharpoonup Y,} f : X Y , {\displaystyle f:X\nrightarrow Y,} or f : X Y . {\displaystyle f:X\hookrightarrow Y.} However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings.

Specifically, for a partial function f : X Y , {\displaystyle f:X\rightharpoonup Y,} and any x X , {\displaystyle x\in X,} one has either:

  • f ( x ) = y Y {\displaystyle f(x)=y\in Y} (it is a single element in Y), or
  • f ( x ) {\displaystyle f(x)} is undefined.

For example, if f {\displaystyle f} is the square root function restricted to the integers

f : Z N , {\displaystyle f:\mathbb {Z} \to \mathbb {N} ,} defined by:
f ( n ) = m {\displaystyle f(n)=m} if, and only if, m 2 = n , {\displaystyle m^{2}=n,} m N , n Z , {\displaystyle m\in \mathbb {N} ,n\in \mathbb {Z} ,}

then f ( n ) {\displaystyle f(n)} is only defined if n {\displaystyle n} is a perfect square (that is, 0 , 1 , 4 , 9 , 16 , {\displaystyle 0,1,4,9,16,\ldots } ). So f ( 25 ) = 5 {\displaystyle f(25)=5} but f ( 26 ) {\displaystyle f(26)} is undefined.