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Thread: uncountability

  1. #1
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    uncountability

    We want to prove that $\displaystyle [a,b] $ and $\displaystyle (a,b) $ are uncountable.

    The book says that there are bijections from $\displaystyle (a,b) $ onto $\displaystyle (-1,1) $ onto unit semicircle. From this point how do we deduce that $\displaystyle (a,b) $ is uncountable?

    $\displaystyle [a,b] $ is then uncountable if we prove that $\displaystyle (a,b) $ is uncountable.
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  2. #2
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    Quote Originally Posted by shilz222 View Post
    We want to prove that $\displaystyle [a,b] $ and $\displaystyle (a,b) $ are uncountable.

    The book says that there are bijections from $\displaystyle (a,b) $ onto $\displaystyle (-1,1) $ onto unit semicircle. From this point how do we deduce that $\displaystyle (a,b) $ is uncountable?

    $\displaystyle [a,b] $ is then uncountable if we prove that $\displaystyle (a,b) $ is uncountable.
    The diagnol argument shows that $\displaystyle (0,1)$ is uncountable. That that means $\displaystyle (-1,1)$ is uncountable because it contains it as a subset.
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