# 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?

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

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 $\mathbb{R}^3$.

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".

• $A\in\mathbb{R}^3$ 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 $\mathbf{a}\in\mathbb{R}^3$? – Clement C. Jun 23 '17 at 16:40
• I updated the question, hope it is clearer. – JDoeDoe Jun 23 '17 at 17:19
• @ClementC. I updated my question. – JDoeDoe Jun 23 '17 at 17:19
• @JDoeDoe See my edit. – Ethan Bolker Jun 23 '17 at 17:34