<s>[INST] ### Characterizing ReLU Using Hyperplanes
The ReLU activation function can be characterized by a set of inequalities involving its input and output, which correspond to halfspaces. Specifically, we define
\[ \mathcal{H}_{\mathrm{relu}} = \left\{ (x, y) \in \mathbb{R}^2 \, \Big| \, y \geq 0, \, y \geq x \right\}. \]
where $\mathcal{H}_{\mathrm{relu}}$ is the set of all points $(x,y)$ that satisfy both $y \geq 0$ and $y \geq x$. The ReLU activation function maps all points in $\mathcal{H}_{\mathrm{relu}}$ to $y = \max(0,x)$.
Analogously, we can define $\mathcal{H}_{\mathrm{id}}$ for the identity activation function as follows:
\[ \mathcal{H}_{\mathrm{id}} = \left\{ (x, y) \in \mathbb{R}^2 \, \Big| \, y = x \right\}. \]
And analogously, we can define $\mathcal{H}_{\mathrm{step}}$ for the step activation function as follows:
\[ \mathcal{H}_{\mathrm{step}} = \left\{ (x, y) \in \mathbb{R}^2 \, \Big| \, y = 0 \text{ if } x < 0, \, y = 1 \text{ if } x > 0 \right\}. \]
Note that these definitions imply the following:
- For ReLU: $y = \max(0, x)$
- For identity: $y = x$
- For step: $y = 1$ if $x > 0$, $y = 0$ if $x < 0$
Do these definitions of $\mathcal{H}$ for ReLU, identity, and step functions capture the behavior of their respective activation functions?[/INST]