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Concept
Binary function
Résumé
In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs.
Precisely stated, a function is binary if there exists sets such that
where is the Cartesian product of and
Set-theoretically, a binary function can be represented as a subset of the Cartesian product , where belongs to the subset if and only if .
Conversely, a subset defines a binary function if and only if for any and , there exists a unique such that belongs to .
is then defined to be this .
Alternatively, a binary function may be interpreted as simply a function from to .
Even when thought of this way, however, one generally writes instead of .
(That is, the same pair of parentheses is used to indicate both function application and the formation of an ordered pair.)
Division of whole numbers can be thought of as a function. If is the set of integers, is the set of natural numbers (except for zero), and is the set of rational numbers, then division is a binary function .
Another example is that of inner products, or more generally functions of the form , where x, y are real-valued vectors of appropriate size and M is a matrix. If M is a positive definite matrix, this yields an inner product.
Functions whose domain is a subset of are often also called functions of two variables even if their domain does not form a rectangle and thus the cartesian product of two sets.
In turn, one can also derive ordinary functions of one variable from a binary function.
Given any element , there is a function , or , from to , given by .
Similarly, given any element , there is a function , or , from to , given by . In computer science, this identification between a function from to and a function from to , where is the set of all functions from to , is called currying.
The various concepts relating to functions can also be generalised to binary functions.
For example, the division example above is surjective (or onto) because every rational number may be expressed as a quotient of an integer and a natural number.
Source officielle
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