# I Sets, Cartesian product

1. May 12, 2017

### control

Hi guys,
I would like to ask if a set can contain coordinates of points, for example A={[1,3];[4,5];[4,7]} and if we can do Cartesian product of such sets, for example A={[1,3];[4,5]}, B={[7,8];[4,2]} A×B={[1,3][7,8];[1,3][4,2];[4,5][7,8];[4,5][4,2]} (is it correct to write it like that?). I am familiar with doing that when we have sets of numbers (A={1;2}, B={7;5} A×B={[1,7];[1,5];[2,7];[2,5]}). but I am not sure if it is correct with coordinates of points.
Mod note: Fixed typo "carthesian"

Last edited by a moderator: May 13, 2017
2. May 12, 2017

### Staff: Mentor

You can write the elements whichever you want, e.g. $[1,3][7,8]$ or $[1,3;7,8]$ or $[1,3,7,8]$ or $\begin{bmatrix}1&3\\7&8\end{bmatrix}$. It is certainly useful not to mix them like $[1,7][3,8]$, because this would probably be harder to read, but as long as you're consistent, there is no rule.
I would probably write $A=\{(1,3),(4,5)\}\; , \;B=\{(7,8),(4,2)\}$ as round parenthesis are more common for tuples and commas as separators in a list, and then $A \times B = \{\; ((1,3),(7,8))\, , \, ((1,3),(4,2))\; , \;((4,5),(4,2))\, , \,((4,5),(4,2))\;\}$ but only in set theory. With different applications, this might change.

3. May 12, 2017

### control

4. May 13, 2017

### zwierz

Perhaps the definition of the Cartesian product would be of some use. Let $\Gamma$ be be an arbitrary nonvoid set, and a set $A_\gamma$ is putted in correspondence to each element $\gamma\in\Gamma$. Then by definition a set $\Pi_{\gamma\in \Gamma}A_\gamma$ consists of functions $f:\Gamma\to \bigcup_{\gamma\in \Gamma}A_{\gamma}$ such that $f(\gamma)\in A_\gamma$.
For example a set $\mathbb{R}\times\mathbb{N}$ consists of functions $f:\{1,2\}\to \mathbb{R}\cup\mathbb{N}$ (it looks little bit strange, obviously $\mathbb{R}\cup\mathbb{N}=\mathbb{R}$) such that $f(1)=a_1\in\mathbb{R},\quad f(2)=a_2\in\mathbb{N}$. This function is also presented as $(a_1,a_2)$.
Another example: a set $\mathbb{R}^\mathbb{N}$ consists of all functions $f:\mathbb{N}\to\mathbb{R}$ those functions can be presented as infinite sequences $(a_1,a_2,\ldots),\quad f(i)=a_i\in\mathbb{R}$.
If all the $A_\gamma$ are vector spaces over the same field then $\Pi_{\gamma\in \Gamma}A_\gamma$ is also a vector space. By the Choice axiom the set $\Pi_{\gamma\in \Gamma}A_\gamma$ is not empty as long as all the sets $A_\gamma$ are not empty

Last edited: May 13, 2017
5. May 13, 2017

### Staff: Mentor

You forgot to mention that the Cartesian product solves a universal mapping problem.