- Binary function
In

mathematics , a**binary function**, or**function of two variables**, is a function which takes two inputs.Precisely stated, a function $f$ is binary if there exists sets $X,\; Y,\; Z$ such that:$,f\; colon\; X\; imes\; Y\; ightarrow\; Z$where $X\; imes\; Y$ is the

Cartesian product of $X$ and $Y.$For example, if

**Z**is the set ofinteger s,**N**^{+}is the set ofnatural number s (except for zero), and**Q**is the set ofrational number s, then division is a binary function from**Z**and**N**^{+}to**Q**.Set-theoretically, one may represent a binary function as a

subset of theCartesian product "X" × "Y" × "Z", where ("x","y","z") belongs to the subsetif and only if "f"("x","y") = "z".Conversely, a subset "R" defines a binary function if and only if,for any "x" in "X" and "y" in "Y",there exists aunique "z" in "Z" such that ("x","y","z") belongs to "R".We then define "f"("x","y") to be this "z".Alternatively, a binary function may be interpreted as simply a function from "X" × "Y" to "Z".Even when thought of this way, however, one generally writes "f"("x","y") instead of "f"(("x","y")).(That is, the same pair of parentheses is used to indicate both function application and the formation of an

ordered pair .)In turn, one can also derive ordinary functions of one variable from a binary function.Given any element "x" of "X", there is a function "f"

^{"x"}, or "f"("x",·), from "Y" to "Z", given by "f"^{"x"}("y") := "f"("x","y").Similarly, given any element "y" of "Y", there is a function "f"_{"y"}, or "f"(·,"y"), from "X" to "Z", given by "f"_{"y"}("x") := "f"("x","y"). (In computer science, this identification between a function from "X" × "Y" to "Z" and a function from "X" to "Z"^{"Y"}is calledCurrying .)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.This example is "injective" in each input separately, because the functions "f"

^{"x"}and "f"_{"y"}are always injective.However, it's not injective in both variables simultaneously, because (for example) "f"(2,4) = "f"(1,2).One can also consider "partial" binary functions, which may be defined only for certain values of the inputs.For example, the division example above may also be interpreted as a partial binary function from

**Z**and**N**to**Q**, where**N**is the set of all natural numbers, including zero.But this function is undefined when the second input is zero.A

binary operation is a binary function where the sets "X", "Y", and "Z" are all equal; binary operations are often used to definealgebraic structure s.In

linear algebra , a bilinear transformation is a binary function where the sets "X", "Y", and "Z" are allvector space s and the derived functions "f"^{"x"}and "f"_{"y"}are alllinear transformation s.A bilinear transformation, like any binary function, can be interpreted as a function from "X" × "Y" to "Z", but this function in general won't be linear.However, the bilinear transformation can also be interpreted as a single linear transformation from thetensor product "X" "Y" to "Z".The concept of binary function generalises to "ternary" (or "3-ary") "function", "quaternary" (or "4-ary") "function", or more generally to "n-ary function" for any

natural number "n".A "0-ary function" to "Z" is simply given by an element of "Z".One can also define an "A-ary function" where "A" is any set; there is one input for each element of "A".In

category theory , "n"-ary functions generalise to "n"-ary morphisms in amulticategory .The interpretation of an "n"-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original "n"-ary morphism will work in amonoidal category .The construction of the derived morphisms of one variable will work in aclosed monoidal category .The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.

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