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Math Help - A paradox

  1. #1
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    A paradox

    I'm sure there is a name for this paradox, but I haven't studied set theory enough to know what it's called. Can someone please help resolve this?

    Let N = the set of the natural numbers, and let D = the set of natural odd numbers.

    N and D have equal cardinalities, and yet N is unequal to D. This is not a problem, for two sets can have similar cardinalities while maintaining their uniqueness.

    The problem, for me anyway, is this: N and D are both infinite sets, and N is a proper superset of D. Thus, shouldn't the cardinality of N be greater than that of D?

    Shouldn't a set always have a cardinality greater than that of its proper subset?
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  2. #2
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    Re: A paradox

    Quote Originally Posted by chuckg1982 View Post
    I'm sure there is a name for this paradox, but I haven't studied set theory enough to know what it's called. Can someone please help resolve this?
    Let N = the set of the natural numbers, and let D = the set of natural odd numbers.
    N and D have equal cardinalities, and yet N is unequal to D. This is not a problem, for two sets can have similar cardinalities while maintaining their uniqueness.
    The problem, for me anyway, is this: N and D are both infinite sets, and N is a proper superset of D. Thus, shouldn't the cardinality of N be greater than that of D?
    Shouldn't a set always have a cardinality greater than that of its proper subset?
    You need to study one-to-one correspondences.
    This will really surprise you.
    There are just as many real numbers in (0,1) as there are on the whole number line.
    In other words, that are equally as many numbers between zero and one as there are real numbers.
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