Real Function of Two Variables/Examples/x + y

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Examples of Real Functions of Two Variables

Let $z$ denote the function defined as:

$z = x + y$

The domain of $z$ is:

$\Dom z = \R \times \R$


Proof

The domain of $z$ is given implicitly and conventionally.

What is meant is:

$z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
$\forall \tuple {x, y} \in S: \map z {x, y} = x + y$

There are no restrictions on either $x$ or $y$ for $\map z {x, y}$ to be defined.

Hence the domain of $z$ is the entire Cartesian plane.

$\blacksquare$


Sources