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Math Help - carthesian product of two closed sets

  1. #1
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    carthesian product of two closed sets

    I have a question, what is the set [0,1]\times[0,1]? What does it contain?

    I know that, the unit interval is [0,1]:=\{x\in\mathbb{R}|0\lex\le 1 but then

    [0,1]\times [0,1]=\{(x,y)|0\le x\le 1,0\le y\le 1|x,y\in\mathbb{R}\} is this true?

    but then we must have [0,1]\times [0,1]\subset \mathbb{R}^2

    I was given the following set A=\{(x,x)|x\in[0,1]\} what is it and what does it contain? is it just a set of open intervals of the real line?
    or is it a diagonal of the unit square?
    Last edited by rayman; September 23rd 2013 at 02:45 AM.
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  2. #2
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    Re: carthesian product of two closed sets

    Hey rayman.

    You are correct with your definition of [0,1] x [0,1].

    As for (x,x) x in [0,1] - Consider all the points that it forms (it will form a straight line with y = x where x is in [0,1]).
    Thanks from rayman
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    Re: carthesian product of two closed sets

    thank you chiro!
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  4. #4
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    Re: carthesian product of two closed sets

    Geometrically, [0, 1]x [0, 1] is the square in the xy-plane with verticex (0, 0), (0, 1), (1, 1), and (1, 0). {(x, x)}, x in [0, 1], is the diagonal from (0, 0) to (1, 1).
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