Notation for List where elements are members of a given Set Is $(a, b, c, d) \in\mathbb{R}$ an adequate way to say to "a, b, c, d is a list of real numbers?"
Or would it be better to say: Given list $(a,b,c,d)$ where $a,b,c,d \in\mathbb{R}$ ?
I am pretty clear on set notation but my search on list notation has been inconclusive.
 A: The latter is standard when the numbers are being considered separately. The former is probably what you want, but you need to be careful. $\mathbb{R}$ is a set, but it doesn't contain any lists - it only contains individual numbers as its elements. 
When you write a list of numbers or symbols in round parentheses, most mathematicians would call that an $n$-tuple. Typically we would write $$(a, b, c, d)\in \mathbb{R}^4$$ 
Here $\mathbb{R}^4$ means the product of four copies of the real numbers; this is the set of all possible $4$-tuples, where each entry is a real number.
Summary: close, but no cigar.
A: In model theory, it is common to write $\overline{x}$ for a list of elements $\langle x_1, \ldots, x_n\rangle$ (or $(x_1, \ldots, x_n)$, if you prefer) and then to write $\overline{x} \in X$ to mean that each $x_i \in X$. Apart from that abuse of notation in model theory, $(a, b, c, d) \in \Bbb{R}$ only makes sense if you have somehow defined $\Bbb{R}$ to be a set of 4-tuples and are talking about the internal details of those 4-tuples (which is almost certainly not the case in normal mathematical writing).
