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I have a question about presenting the set of point in the math. I have two sets in domain $\Omega$. $M$ is set of $m$ point locations, $N$ is set of $n$ point locations. In which,some points in the set $M$ have same location with set $N$. I want to represent some condition in mathematical way. Could you help me represent it?

  1. Represent set $M,N$
  2. Represent the intersection between $M$ and $N$
  3. Represent points $\in M$ but not belong $N$

Thank you in advance.

These are my results. But I am not sure about my answers.

  1. Represent set $M,N$ $$M=\{(x_i,y_i) \mid i=1,\dots,m,(x_i,y_i) \in \Omega \}$$ (Or only $M=\{(x,y) \mid (x,y) \in \Omega \}$???) $$N=\{(x_j,y_j) \mid j=1,\dots,n, (x_n,y_n) \in \Omega \}$$
  2. Represent the intersection between $M$ and $N$ $$K=M\bigcap N=\{(x_k,y_k) \mid (x_k,y_k) \in \Omega, (x_k,y_k) \in M \text{ and } (x_k,y_k) \in N \}$$
  3. Represent points $\in M$ but not belong $N$ $$H=\{(x_h,y_h) \mid (x_h,y_h) \in \Omega, (x_h,y_h) \in M \text{ and } (x_h,y_h) \notin N \}$$
john2182
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    Representation theory is something else. Wrong tag. – user4894 Mar 13 '15 at 18:00
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    Is the domain $\Omega$ a subset of $\Bbb{C}$ or $\Bbb{R}^2$? If so, then you should say so. It seems like you want to describe the points in $\Omega$ as ordered pairs, so I can guess that $\Omega \subseteq \Bbb{R}^2$. – Sammy Black Mar 13 '15 at 18:08
  • I don't really know what you could say, other than $M$ is larger than $N$ if $m \geq n$. – William Stagner Mar 13 '15 at 18:10
  • @SammyBlack: Yes. It \in $R^2$. For example, it is image domain. William Stagner: Because I don't know the relationship between $m$ and $n$. So I write "we don't know which is bigger". Sometime, m is bigger than n.But sometime m is smaller than n – john2182 Mar 13 '15 at 18:15
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    Then it's not really clear what you're asking. – William Stagner Mar 13 '15 at 18:24
  • I must represent condition 1 to 3 by mathematic. That is my question – john2182 Mar 13 '15 at 18:26
  • @ William Stagner: I updated my question. I deleted the sentence $m>n$ – john2182 Mar 13 '15 at 19:23

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