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Set theory & equiv classes

  1. Aug 13, 2009 #1
    How do you write in proper set theoretic notation that
    a set A = (x,x) where x is a non-negative real number?

    Also, (x_1, y_1) R (x_2, y_2) if x_1 ^2 + y_1 ^2 = x_2 ^2 + y_2 ^2
    The equiv. classes are circles at (0,0), right?
    How do you write this formally (using set theoretic notation)?
     
  2. jcsd
  3. Aug 13, 2009 #2
    If it's clear from the context that you're working in the real numbers with the standard ordering, then you'd probably write [tex]A=\{(x,x)|x>0\}[/tex]. Otherwise, you'd usually write either [tex]A=\{(x,x)|x\in\mathbb{R}^+_0\}[/tex] or [tex]A=\{(x,x)|x\in[0,\infty)\}[/tex], although these are by no means the only conventions.

    In general, to define a set using set notation, you need to specify (a) a collection of variables, (b) what condition those variables must satisfy, and (c) how the variables are combined to make an object in the overall set. In other words, a set S is given in set notation by
    [tex]S=\{L(x_1,x_2,\ldots)|Q(x_1,x_2,\ldots)\}[/tex]​
    where Q is some condition (e.g. the condition that [tex]x_1[/tex] is a blue ball, [tex]x_2[/tex] is a real number, and all the other variables are stars in the Andromeda galaxy) and L specifies a way of combining variables.
     
  4. Aug 13, 2009 #3
    Yes, they are. Formally, the equivalence class [x,y] of (x,y) is
    [tex][x,y]=\left\{(x',y')|x'^2+y'^2=x^2+y^2\right\}.[/tex]​
     
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