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Cardinality of infinite subset of infinite set

  1. Dec 6, 2013 #1
    Am a bit confused about the meaning of cardinality. If ## A \subseteq B ##, then is it necessarily the case that ## |A| \leq |B| ##?

    I am thinking that since ## A \subseteq B ##, an injection from A to B exists, hence its cardinality cannot be greater than that of B?

    But this cannot be correct, since ##\mathbb{Z}## and ##\mathbb{Q}## have the same cardinality?

    Where am I wrong?

    Thanks!

    BiP
     
  2. jcsd
  3. Dec 6, 2013 #2
    You are right. If ##A\subseteq B##, then ##|A|\leq |B|##.

    In particular, ##|\mathbb{Z}|\leq |\mathbb{Q}|## is true. Don't confuse this with ##|\mathbb{Z}|< |\mathbb{Q}|##, which is false.
     
  4. Dec 7, 2013 #3
    I see. So which of the following is true?
    ##|\mathbb{Z}|< |\mathbb{Q}|##
    ##|\mathbb{Z}|= |\mathbb{Q}|##

    Thanks!

    BiP
     
  5. Dec 7, 2013 #4

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