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I Sets, Cartesian product

  1. May 12, 2017 #1
    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. jcsd
  3. May 12, 2017 #2

    fresh_42

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    2017 Award

    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.
     
  4. May 12, 2017 #3
    Thanks for answer.
     
  5. May 13, 2017 #4
    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
  6. May 13, 2017 #5

    fresh_42

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    Staff: Mentor

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