Is there any notation for representing a set of values in a vector ranging from minimum to maximum with incremental step size?

Question

I want to represent the following in set form: Let $\textbf{a}_{(l)}$ represent a vector of $N_a$ distinct attenuation values ranging from minimum value $a_{(l)}^{\min}$ to maximum value $a_{(l)}^{\max}$ with incremental step size of $\Delta a_{(l)}$.

Answer 1

For simplicity of notation, I will call the min value $a$, max value $b$, and step size $h$. $$\{a+hk: 0\leq k \leq ?, k \in \mathbb{Z} \}$$

To solve for ?, we want to solve $a+?h = b$, so $?=\frac{b-a}{h}$. Thus your desired set is

$$\{a+hk: 0\leq k \leq \frac{b-a}{h}, k \in \mathbb{Z} \}.$$

Te be clear this says the set of all $a+hk$ where $k$ takes values between $0$ and $\frac{b-a}{h}$ and $k$ must be an integer. That is the literal way to translate it, but a more eloquent phrasing would be "the set of all $a+hk$ where $k$ is an integer between $0$ and $\frac{b-a}{h}$.

Alternatively:

$$\{a+hk: 0\leq k \leq N_a-1, k \in \mathbb{Z} \}.$$

Since there are $N_a$ values in the vector and this makes $k$ range over $N_a$ different values.