Let's say I have a Heaviside function defined like this:
$$ H(x) = \begin{cases} 0, \text{ if } x < 0\\ 1, \text{ if } x \geq 0 \end{cases} $$
Then I have a so called Iverson brackets:
$$ [condition] = \begin{cases} 1, \text{ if the condition is TRUE} \\ 0, \text{ if the condition is FALSE} \end{cases} $$
I know I can rewrite some cases of Iverson bracket to Heaviside like this:
\begin{align} [x \geq a]&= [x-a \geq 0] = H(x-a) \\ [x < a] &= [\neg(x \geq a)] = [\neg(x-a \geq 0)] = 1-[x - a \geq 0] = 1-H(x-a) \end{align}
But, the question is - how can I rewrite to Heaviside the other two cases $x -a > 0$ and $x-a \leq 0$? They doesn't correspond to Heaviside func. definition and I can't use the simple negation to change them to $x -a \geq 0$ form.
$$[x\le a]=[x-a\le0]=H(a-x)$$$$[x\gt a]=[x-a\gt0]=1-H(a-x)$$