1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted