I already know that the below sentences are equivalent. $U$ is universal set.
$$\forall{x} \in U : q(x) \equiv \forall{x} : [x \in U \rightarrow q(x)]$$
I think that the "$\forall{x}$" in the right sentence is meaningless because it doesn't specify which set the "x" belongs to. So it should be "$\forall{x} \in U$". Is it right? If my logic is right, why do many authors often omit which set the element is in?
In first order logic, sentences are defined in a first order language $\mathcal{L}$, where $x_i$ are called variables and always belong to $\mathcal{L}$. When you talk about $x$ belonging to a set, you are talking about the interpretation of $\mathcal{L}$ in some structure $\mathcal{M}$, where $x$ must belong to a specific set $M$ which itself belongs to $\mathcal{M}$.
For example, $$\forall{x}\forall{y}(x=y)$$ is a sentence in some language $\mathcal{L}$, and talking about its truth is meaningless. More precisely, it may be true or false only when it is interpreted in structures. For instance if $M$ has only one member, then $\forall{x}\forall{y}(x=y)$ is true. However, if $M$ has more than one member, it is false.