• Agent Smith
    9.5k
    First the ...

    Rules

    1. Set A and set B have the same cardinality IFF the elements of set A can be put in a 1-to-1 correspondence with the elements of set B. for example {s, 7, h} can be put in a 1-to-1 correspondence with the set {5, z, %} and so the two have the same cardinality.

    2. Set A has a smaller cardinality than set B IFF set A can be put in a 1-to-1 correspondence with a proper subset of set B. For example, the set {2, g, &} can be put in 1-to-1 correspondence with the proper subset, the set {\, k, f} of the set {p, f, \, k} and so the cardinality of {2, g, &} is less than the cardinality of {p, f, \, k}.

    Then the ...

    Notes:
    a) elements of a set A possess a 1-to-1 correspondence with the elements of a set B IFF each element of set A is matched with exactly one element of set B.

    b) Set A is a proper subset of set B if set B contains all the elements of set A and has at least one element that is not in set A. For example, {1,$} is a proper subset of {d, $, 1}

    c) The cardinality of a set A, n(A), is the number of elements in set A. If A = {e, ¥, 3} then n(A) = 3.

    Lastly, the argument ...

    N = The set of natural numbers = {1, 2, 3, ...}
    E = The set of even numbers = {2, 4, 6, ...}
    O = The set of odd numbers = {1, 3, 5, ...}

    1. Rule 1 implies E has the same cardinality as N: (1, 2), (2, 4), ..., (n, 2n).

    2. Rule 2 implies E's cardinality is less than N's cardinality as E can be put in a 1-to-1 correspondence with O [like so (1, 2), (3, 4), ..., (n, n + 1)] and O is a proper subset of N.

    What gives?
  • bongo fury
    1.6k
    2. Set A has a smaller cardinality than set B IFF set A can be put in a 1-to-1 correspondence with a proper subset of set B.Agent Smith

    No. https://en.wikipedia.org/wiki/Cardinality?wprov=sfla1
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