Editing Binary function (section)

==Alternative definitions==
[[Naive set theory|Set-theoretically]], a binary function can be represented as a [[subset]] of the [[Cartesian product]] <math>X \times Y \times Z</math>, where <math>(x,y,z)</math> belongs to the subset [[if and only if]] <math>f(x,y) = z</math>.
Conversely, a subset <math>R</math> defines a binary function if and only if [[universal quantification|for any]] <math>x \in X</math> and <math>y \in Y</math>, [[existential quantification|there exists]] a [[uniqueness quantification|unique]]  <math>z \in Z</math> such that <math>(x,y,z)</math> belongs to <math>R</math>.
<math>f(x,y)</math> is then defined to be this <math>z</math>.

Alternatively, a binary function may be interpreted as simply a [[function (mathematics)|function]] from <math>X \times Y</math> to <math>Z</math>.
Even when thought of this way, however, one generally writes <math>f(x,y)</math> instead of <math>f((x,y))</math>.
(That is, the same pair of parentheses is used to indicate both [[function application]] and the formation of an [[ordered pair]].)