Translating into predicate logical Every odd number is a sum of two even numbers.
The above translates to $∀_{x∈ℤ}$ $∃_{a,b∈ℤ}$ such that $2x+1=2a+2b$.
The negation of the above is $∃_{x∈ℤ}$ $∀_{a,b∈ℤ}$ such that $2x+1≠2a+2b$
The negation is true, but the original statement is false.
Is my translation valid?
 A: You have a working translation but there are a few notes I might add.
First, your sentences are not entirely in predicate logic, they are still using the English convention "such that" and they are using restricted quantifiers. These are phrases like $\forall x\in\mathbb{Z}$ as opposed to $\forall x$. They are not a part of the language of predicate logic, it is a linguistic convention in mathematics.
Your class might accept those conventions. In case they don't, I'll describe how to work around them.
Removing "such that" is easy. If you have a phrase "$\forall x$ such that $\phi(x)$, it is the case that $\psi(x)$", what this is doing is putting a restriction on $x$ before asserting that $\psi(x)$. In other words, if an $x$ is $\phi$ then we have to have that $x$ is $\psi$. But, we can't deduce that $x$ is $\psi$ unless we have $x$ is $\phi$. This leads the translation $\forall x(\phi(x)\rightarrow \psi(x))$.
Restricted quanitfers are doing the same thing. $\forall x\in\mathbb{Z}\psi(x)$ asserts that if $x\in\mathbb{Z}$, then $\psi(x)$. This is the previous case but $\phi\equiv x\in\mathbb{Z}$ and, well,  $\psi\equiv\psi$.
