What does the notation $x^+$ and $x^-$ mean?
I have been reading through this document, and on the bottom of page 7 it uses the notation
$x(A) = x^+(A) - x^-(A)$
where $x$ is a function.
What could this notation mean?
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$\begingroup$For a function $f(x)$, $f^{+}(x)$ is called the "positive part of the function". It is defined as $$f^{+}(x) := \begin{cases}f(x) & f(x) > 0 \\ 0 & f(x) < 0 \end{cases}$$ showing that it is quite literally the outputs of $f(x)$ that are positive. Similarly, the negative part is defined as $$f^{-}(x) := \begin{cases} -f(x) & f(x) < 0 \\ 0 & f(x) \geq 0 \end{cases}.$$ This shows you that $f^{-}(x)$ is the positive version of the values of $f(x)$ that are negative. The reason to make it the positive version is so that we can write $f(x) = f^{+}(x) - f^{-}(x)$.
A more concise way of defining $f^{+}$ and $f^{-}$, which is equivalent to the above definitions, is $f^{+}(x) := \max\{f(x),0\}$ and $f^{-}(x) := \max\{-f(x),0\}$.
$\endgroup$ $\begingroup$$x^+(A) = \max\{x(A),0\}$ and $x^-(A) = \min\{x(A),0\},$ for a function $x=x(A).$
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