Question: I need a set-builder notation for $\{1 , 3 , 5 , -7 , -9 , -11 , 13 , 15 , 17 , \ldots\}$. It is an infinite set of odd positive integers, which are multiplied by $-1$ three by three. One of the terms of this question is not to use floor function.

Answer: 
Well, no doubt you are hoping for something cleverer than this, but:

$$ \{\,a\mid\exists k\in \mathbb {N}\_{0}:(\exists m \in \{1,3,5\}
:a = m+12k)\vee (\exists m \in \{-7,-9,-11\}
:a = m-12k)\,\}$$ 

which I suddenly realized is actually

$$ \{\,a\mid\exists k\in \mathbb {Z}:(\exists m \in \{1,3,5\}
:a = m+12k)\,\}$$
Define $f:\mathbb{N} \to \mathbb{Z}$ as follows. $f(1) = 1$, $f(2) = 3$ $f(3) = 5$ and $$f(n) = -\text{sgn}(f(n-3))\cdot(2n-1)$$

for $n \geq 4$

Then this set is $\{c \in \mathbb{Z}: \exists n \in \mathbb{N} \ \text{such that} \ f(n) = c\}$. That is, the set is the image of the function.