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Uncountable Union

Rating: 4.5
November 21st, 2007

easyriddle.gifA very interesting riddle for those of you with some basic background in Set Theory.

Prove or Disprove the Following Claim:

There exists a subset B of P(N), such that |B| = A, and B is completely ordered by the subset relation.

Notes:

N denotes the set of natural numbers.

A denotes the cardinality of the continuum (i.e. A = aleph = 2^aleph 0).

“B is completely ordered by the subset relation” means that for every two elements a, b of B, either a is a subset of b or b is a subset of a.

Pages: 1 2

This entry was posted on Wednesday, November 21st, 2007 at 6:09 pm and is filed under Riddles, Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “Uncountable Union”

  1. Nir Lev Says:
    February 16th, 2008 at 4:48 am

    Choose an enumeration of the rational numbers Q = { q(1), q(2), q(3), … }. For any real number x define the set E(x) = {n : q(n)

  2. yaniv Says:
    February 16th, 2008 at 5:40 pm

    Hi Nir,

    I am sorry your comment got chopped. If you want to use the < symbol in a comment you must encode it as the four letter sequence &lt;

    Second, please post solutions to the second page of riddles (as not to ruin them for other people).

    And finally - of course your solution is correct! ;-)

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