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 Zwhere X imes Y is the Cartesian product of X and Y.

For example, if Z is the set of integers, N+ is the set of natural numbers (except for zero), and Q is the set of rational numbers, then division is a binary function from Z and N+ to Q.

Set-theoretically, one may represent a binary function as a subset of the Cartesian product "X" × "Y" × "Z", where ("x","y","z") belongs to the subset if 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 a unique "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 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.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 define algebraic structures.

In linear algebra, a bilinear transformation is a binary function where the sets "X", "Y", and "Z" are all vector spaces and the derived functions "f""x" and "f""y" are all linear transformations.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 the tensor 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 a multicategory.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 a monoidal category.The construction of the derived morphisms of one variable will work in a closed monoidal category.The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • binary function — noun a function of the type , or more generally, a function of the type in which exactly two of the sets U are equal to V …   Wiktionary

  • Binary — means composed of two parts or two pieces . It contrasts with Unary, Ternary, Quaternary, and so on.Binary may also refer to:* Binary option, also known as digital option OR all or nothing option * Binary numeral system, a representation for… …   Wikipedia

  • Binary operation — Not to be confused with Bitwise operation. In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction,… …   Wikipedia

  • Binary — Bi na*ry, a. [L. binarius, fr. bini two by two, two at a time, fr. root of bis twice; akin to E. two: cf. F. binaire.] Compounded or consisting of two things or parts; characterized by two (things). [1913 Webster] {Binary arithmetic}, that in… …   The Collaborative International Dictionary of English

  • Binary arithmetic — Binary Bi na*ry, a. [L. binarius, fr. bini two by two, two at a time, fr. root of bis twice; akin to E. two: cf. F. binaire.] Compounded or consisting of two things or parts; characterized by two (things). [1913 Webster] {Binary arithmetic}, that …   The Collaborative International Dictionary of English

  • Binary compound — Binary Bi na*ry, a. [L. binarius, fr. bini two by two, two at a time, fr. root of bis twice; akin to E. two: cf. F. binaire.] Compounded or consisting of two things or parts; characterized by two (things). [1913 Webster] {Binary arithmetic}, that …   The Collaborative International Dictionary of English

  • Binary logarithms — Binary Bi na*ry, a. [L. binarius, fr. bini two by two, two at a time, fr. root of bis twice; akin to E. two: cf. F. binaire.] Compounded or consisting of two things or parts; characterized by two (things). [1913 Webster] {Binary arithmetic}, that …   The Collaborative International Dictionary of English

  • Binary measure — Binary Bi na*ry, a. [L. binarius, fr. bini two by two, two at a time, fr. root of bis twice; akin to E. two: cf. F. binaire.] Compounded or consisting of two things or parts; characterized by two (things). [1913 Webster] {Binary arithmetic}, that …   The Collaborative International Dictionary of English

  • Binary nomenclature — Binary Bi na*ry, a. [L. binarius, fr. bini two by two, two at a time, fr. root of bis twice; akin to E. two: cf. F. binaire.] Compounded or consisting of two things or parts; characterized by two (things). [1913 Webster] {Binary arithmetic}, that …   The Collaborative International Dictionary of English

  • Binary scale — Binary Bi na*ry, a. [L. binarius, fr. bini two by two, two at a time, fr. root of bis twice; akin to E. two: cf. F. binaire.] Compounded or consisting of two things or parts; characterized by two (things). [1913 Webster] {Binary arithmetic}, that …   The Collaborative International Dictionary of English

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”