Alternative Definitions
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.)
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