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Discrete math - Infinite sets having the same cardinality.

  1. Oct 23, 2011 #1
    From a pdf textbook:
    Example (infinite sets having the same cardinality). Let f : (0, 1) → (1,∞) be
    defined by f(x) = 1/x. Then f is a 1-1 correspondence. (Exercise: prove it.) Therefore,
    |(0, 1)| = |(1,∞)|.

    Exercise. Show that |(0,∞)| = |(1,∞)| = |(0, 1)|. Use this result and the fact that
    (0,∞) = (0, 1) ∪ {1} ∪ (1,∞) to show that |(0, 1)| = |R|.


    This example greatly confused me. A 1-1 correspondence (aka a bijection) needs a unique value in the domain for each value in the range. There are only two values, (0,1), that map to Z+, so how can it be a 1-1 correspondence?

    I also do not understand what the function f(x) = 1/x would look like with only (0,1) as its domain. Could someone expand what this would look like?

    EDIT: solved. I understand now. (0,1) is an interval containing all decimal values between 0 and 1. I thought this question was asking for an infinite string of binary values or something like that.
     
    Last edited: Oct 23, 2011
  2. jcsd
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