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(A function of arity n thus has arity n+1 considered as a relation.) In computer programming, there is often a syntactical distinction between operators and functions; syntactical operators usually have arity 1, 2, or 3 (the ternary operator?: is also common). Functions vary widely in the number of arguments, though large numbers can become ...
In mathematics and optimization, a pseudo-Boolean function is a function of the form :, where B = {0, 1} is a Boolean domain and n is a nonnegative integer called the arity of the function. A Boolean function is then a special case, where the values are also restricted to 0 or 1.
In a sense, these are nullary (i.e. 0-arity) predicates. In first-order logic, a predicate forms an atomic formula when applied to an appropriate number of terms. In set theory with the law of excluded middle, predicates are understood to be characteristic functions or set indicator functions (i.e., functions from a set element to a truth value).
The hypergeometric function is an example of a four-argument function. The number of arguments that a function takes is called the arity of the function. A function that takes a single argument as input, such as f ( x ) = x 2 {\displaystyle f(x)=x^{2}} , is called a unary function .
An n-ary operation ω on a set X is a function ω: X n → X. The set X n is called the domain of the operation, the output set is called the codomain of the operation, and the fixed non-negative integer n (the number of operands) is called the arity of the operation. Thus a unary operation has arity one, and a binary operation has arity two.
The practical motivation for partial application is that very often the functions obtained by supplying some but not all of the arguments to a function are useful; for example, many languages have a function or operator similar to plus_one. Partial application makes it easy to define these functions, for example by creating a function that ...
The interpretation of a constant symbol (a function symbol of arity 0) is a function from D 0 (a set whose only member is the empty tuple) to D, which can be simply identified with an object in D. For example, an interpretation may assign the value I ( c ) = 10 {\displaystyle I(c)=10} to the constant symbol c {\displaystyle c} .
In contrast, partial function application refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given the definition of f {\displaystyle f} above, we might fix (or 'bind') the first argument, producing a function of type partial ( f ) : ( Y × Z ) → N {\displaystyle {\text{partial ...