# Notation of the set of all the numbers $y$ satisfying an equation $y= ax + b$ with $x \in \mathbb{Z}$

I want to know the notation of a set $\mathcal{S}$ containing all $y$ that satisfy an equation $y = ax + b$ with $x \in \mathbb{Z}$, for example $2x - 5$ or $\pi x + \frac {5}{2}\pi$.

For instance, given the equation $y = 2x + 3$ ($x \in \mathbb{Z}$), the set of solutions would be $\mathcal{S} = \{\dots , 1, 3, 5, \dots \}$.

• How about $a\mathbb Z + b$? Aug 10, 2016 at 21:31
• So $\mathcal{S} = a\mathbb{Z} + b$ or $\mathcal{S} = \{ y | y = a\mathbb{Z} + b \}$? Aug 10, 2016 at 21:34
• $a\mathbb Z + b = \{ ax+b \mid x\in \mathbb Z \}$ Aug 10, 2016 at 21:38

the notation is simple. in general: $$\mathcal S = \{\textrm{element} | \textrm{constraints to the element}\}$$ in this case you could write (most used notation): $$\mathcal S = \{ax+b| x \in \mathbb Z\}$$ or equivalently: $$\mathcal S = \{y \in \mathbb R|y=ax+b, x \in \mathbb Z\}$$ or for the most compactness: $$\mathcal S = a \mathbb Z+b$$
The response $a\mathbb{Z}+b$, given by others, is certainly correct. It is worth noticing, though, that this designation is not unique: if $b_1$ and $b_2$ are two numbers that differ only by a multiple of $a$, then $a\mathbb{Z}+b_1$ and $a\mathbb{Z}+b_2$ describe the exact same set.
• As $2\mathbb{Z}+3$
• As $2\mathbb{Z}+1$
• As $2\mathbb{Z}-1$
• or indeed as $2\mathbb{Z}+n$ for any odd number $n$.