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?
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$\begingroup$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 $\,\mathbf{U} \times \mathbf{V} \times \mathbf{W}=\mathbb{R}^3\,$.
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".
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