I'm currently working on some discrete mathematics work and I've encountered a question I'm not sure how to answer exactly. Precisely, I'm trying to translate or re-write a division algorithm into symbolic logic and my understanding of how to do that doesn't seem to be enough. The theorem in question goes as follows:
Write the following theorem in symbolic logic:
For any integers a,b ∈ Z, with b > 0, there exist unique integers
q and r such that
a = qb + r with 0 <= r < b
Should I make some form of truth table to better understand how this could be translated into symbolic logic? Any help is appreciated.
*Edit2: So could it be rewritten such that
∀ a, b ∈ Z, where b > 0, ∃! q,r ∈ Z such that a = qb + r with 0 <= r < b
or is this a misunderstanding on my part?
$$ \forall (a, b)\in\mathbb{Z},\exists!(q, r)\in\mathbb{Z}\ :\left((b\gt 0)\to\left(0\leq r\lt b \land a=qb+r\right)\right)$$