Definition:Extended Real Subtraction

Definition
Let $\overline{\R}$ denote the extended real numbers.

Define extended real subtraction or subtraction on $\overline{\R}$, denoted $-_{\overline{\R}}: \overline{\R} \times \overline{\R} \to \overline{\R}$, by:


 * $\forall x,y \in \R: x -_{\overline{\R}} y := x -_{\R} y$ where $-_\R$ denotes real subtraction
 * $\forall x \in \R: x -_{\overline{\R}} \left({+\infty}\right) = \left({-\infty}\right) -_{\overline{\R}} x := -\infty$
 * $\forall x \in \R: x -_{\overline{\R}} \left({-\infty}\right) = \left({+\infty}\right) -_{\overline{\R}} x := +\infty$
 * $\left({-\infty}\right) -_{\overline{\R}} \left({+\infty}\right) := -\infty$
 * $\left({+\infty}\right) -_{\overline{\R}} \left({-\infty}\right) := +\infty$

In particular, the expressions:


 * $\left({+\infty}\right) -_{\overline{\R}} \left({+\infty}\right)$
 * $\left({-\infty}\right) -_{\overline{\R}} \left({-\infty}\right)$

are considered void and should be avoided.

When no danger of confusion arises, $-_{\overline{\R}}$ is usually replaced with the more familiar $-$.

From the definition of $-_{\overline{\R}}$ on bona fide real numbers, the name extended real subtraction is appropriate: the real subtraction is indeed extended.

Caution
While it is tempting to think of extended real subtraction as simply subtraction, there are some intricacies:
 * It is not the case that $- \left({+\infty}\right) = -\infty$ in the sense of additive inverse, because $\left({+\infty}\right) +_{\overline{\R}} \left({-\infty}\right)$ is not defined, and in particular, not equal to $0$.
 * $-_{\overline{\R}}$ is not a mapping as it isn't defined on all of $\overline{\R} \times \overline{\R}$; however, it is a partial mapping