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To be specific, suppose X, Y, and Z are sets. Suppose that, given any elements x of X and y of Y, f(x,y) is a unique element of Z. Then f is a binary function from X and Y to Z.

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.

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 pairAn ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element''. An ordered pair with first element a and second element b is usually written as a b . The notation a b is also us.)

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).

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 injectiveIn mathematics, an injective function (or one-to-one function or injection is a function which maps distinct input values to distinct output values. This is in contrast to a "many-to-one" function, which may map two distinct input values to the same outpu 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 operationIn mathematics, a binary operation or binary operator is a calculation involving two input quantities and one kind of a specific operation. It is sometimes called a dyadic operation as well. Examples include the familiar arithmetic operations of addition, is a binary function where the sets X, Y, and Z are all equal; binary operations are often used to define algebraic structureIn abstract algebra, an algebraic structure consists of a set together with one or more operations on the set which satisfy certain axioms. In case there are no ambiguities, we usually identify the set with the algebraic structure. For example, a group Gs.

In linear algebraLinear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is wi, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (fors 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.