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A set-valued function, also called a correspondence or set-valued relation, is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. [ 1 ] [ 2 ] Set-valued functions are used in a variety of mathematical fields, including optimization , control theory and game theory .
Sigma function: Sums of powers of divisors of a given natural number. Euler's totient function: Number of numbers coprime to (and not bigger than) a given one. Prime-counting function: Number of primes less than or equal to a given number. Partition function: Order-independent count of ways to write a given positive integer as a sum of positive ...
A dfn can be anonymous; a tradfn must be named. A dfn is named by assignment (←); a tradfn is named by embedding the name in the representation of the function and applying ⎕fx (a system function) to that representation. A dfn is handier than a tradfn as an operand (see preceding items: a tradfn must be named; a tradfn is named by embedding
A bijective function is also called a bijection or a one-to-one correspondence (not to be confused with one-to-one function, which refers to injection). A function is bijective if and only if every possible image is mapped to by exactly one argument. [1] This equivalent condition is formally expressed as follows:
Codomain – Target set of a mathematical function; Range of a function – Subset of a function's codomain; Image (mathematics) – Set of the values of a function; Injective function – Function that preserves distinctness; Surjection – Mathematical function such that every output has at least one input; Bijection – One-to-one correspondence
In computer programming, a naming convention is a set of rules for choosing the character sequence to be used for identifiers which denote variables, types, functions, and other entities in source code and documentation. Reasons for using a naming convention (as opposed to allowing programmers to choose any character sequence) include the ...
However, when a value is assigned to x, such as lava, the function then has the value true; while one assigns to x a value like ice, the function then has the value false. Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrote in The Principles of Mathematics (page 106):
Given two sets and , let be a multivalued map from to (equivalently, : is a function from to the power set of ).. A function : is said to be a selection of , if: (() ()).The existence of more regular choice functions, namely continuous or measurable selections is important in the theory of differential inclusions, optimal control, and mathematical economics. [2]