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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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  • $\begingroup$ That $A$ you have written is not a vector, it is a set of vectors. $\endgroup$ – GEdgar Jun 23 '17 at 16:59
  • $\begingroup$ It is sufficient to just write $\mathbf a\in\mathbb R^3$, that is all. $\endgroup$ – Rahul Jun 23 '17 at 17:22
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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 $\,\mathbf{U} \times \mathbf{V} \times \mathbf{W}=\mathbb{R}^3\,$.

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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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  • $\begingroup$ $A\in\mathbb{R}^3$ means $A$ is a vector, not a set of all vectors in $\mathbb{R}^3$. I am not clear about what the OP wants to do, but maybe you wanted to write $\mathbf{a}\in\mathbb{R}^3$? $\endgroup$ – Clement C. Jun 23 '17 at 16:40
  • $\begingroup$ I updated the question, hope it is clearer. $\endgroup$ – JDoeDoe Jun 23 '17 at 17:19
  • $\begingroup$ @ClementC. I updated my question. $\endgroup$ – JDoeDoe Jun 23 '17 at 17:19
  • $\begingroup$ @JDoeDoe See my edit. $\endgroup$ – Ethan Bolker Jun 23 '17 at 17:34

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