How to write a vector with set notation? I have a vector $\mathbf{a}=(a_1,a_2,a_3)$, where $a_1,a_2,a_3$ are real numbers. 
I now want to write that $\mathbf{a}$ is a vector in $\mathbb{R}^3$ and that $a_1,a_2,a_3$ are real numbers. What is the proper notation for this?
Is it correct to write
$$
A=\big\{\mathbf{a}=(a_1,a_2,a_3)
\in\mathbb{R}^3:a_1\in\mathbb{R}, a_2\in\mathbb{R} \text{ and } a_3\in\mathbb{R}
\big\} \quad \text{?}
$$
Or something else?
 A: 
I now want to write that $\mathbf{a}$ is a vector in $\mathbb{R}^3$ and that $a_1,a_2,a_3$ are real numbers.

All you need to write is $\;\mathbf{a}=(a_1,a_2,a_3) \in \mathbb{R}^3\,$.

Is it correct to write
  $$
A=\big\{\mathbf{a}=(a_1,a_2,a_3)
\in\mathbb{R}^3:a_1\in\mathbb{R}, a_2\in\mathbb{R} \text{ and } a_3\in\mathbb{R}
\big\} \quad \text{?}
$$



*

*The above does not define one vector, but a set of vectors. In fact, the way it's written $A=\mathbb{R}^3$.

*$a_1\in\mathbb{R} \dots$ is redundant. When you write $(a_1,a_2,a_3)
\in\mathbb{R}^3$ this implies $a_1, a_2, a_3 \in\mathbb{R}\,$. More generally, when you write $(a_1,a_2,a_3)
\in\mathbf{U} \times \mathbf{V} \times \mathbf{W}$ this implies $a_1\in\mathbf{U}\,$, $a_2\in\mathbf{V}\,$, $a_3\in\mathbf{W}\,$. In the case here $\,\mathbf{U}=\mathbf{V}=\mathbf{W}=\mathbb{R}\,$, so $\,\mathbf{U} \times \mathbf{V} \times \mathbf{W}=\mathbb{R}^3\,$.
A: Edit in response to edited question.
All you have to say is

$a$ is a vector in $\mathbb{R}^3$.

That tells your reader there are three real coordinates.

What you've written is literally correct, but weird. Your set $A$ is nothing but $\mathbb{R}^3$, which you've used in the definition of $A$.

So you can (and should) just write
$$
a \in \mathbb{R}^3
$$
(You need $\in$, not $\subset$).
So it's not clear what you are trying to accomplish with "set notation".
