# Two-Input Function

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## References

### 2011

• http://en.wikipedia.org/wiki/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 $\,f \colon X \times Y \rightarrow Z$ where $X \times Y$ 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.)