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Math Help - find cardinatliy

  1. #1
    Member Jskid's Avatar
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    find cardinatliy

    Let S be the set of all real numbers in the interval (0, 1) whose decimal expansions contain only 0's, 2's and 7's. Let S' be the elements of S whose decimal expansions contain only finitely many 2's and 7's. What is the cardinality of S'?

    I think it is \aleph_0 because even if it contains finitely 2's and 7's you could still make infinity numbers, or am I wrong? For example you could always tack on another 2 or 7 to make a new number, which still has a finite number of digits.
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  2. #2
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    Re: find cardinatliy

    Yes, S' is infinite. On the other hand, all numbers in S' are rational, so it is countable. So, you are right, the cardinality of S' is \aleph_0.
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  3. #3
    Member Jskid's Avatar
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    Re: find cardinatliy

    In these types of questions how do you show your work?
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    Re: find cardinatliy

    Quote Originally Posted by Jskid View Post
    In these types of questions how do you show your work?
    That is a question that only your instructor/tutor can answer fully.
    That said, if I were you, I would review my notes to find theorems and other problems of a similar nature.

    For example, have you shown that set of all finite subsets of \mathbb{N} is countable?
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