Definable sets in $(\mathbb{Z}, +)$

Question

I want to proove that set of all odd numbers is definable in $(\mathbb{Z}, +)$.

I am thinking about next formula $$ \varphi{(y)} \equiv \exists x \;(x + x + 1 = y) $$ I think $1$ is not definable element in $(\mathbb{Z}, +)$. So, I don't know how to solve the problem.

The same thing is with the set whose numbers are of the form $4k+2$ and $6k+3$.

Thanks a lot

Answer 1

For odd numbers: How about trying to define the even numbers, and then negate the formula?

For $4k+2$: Numbers that are not divisible by $4$ but are even.

For $6k+3$: Numbers that are not divisible by $6$ but are divisible by $3$.