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Math Help - Set Theory: that squirrelly 'empty set'

  1. #1
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    Set Theory: that squirrelly 'empty set'

    Okay, I'm new to set theory and having a little trouble wrapping my head around how empty set works.

    (I'll represent empty set with the unicode character "∅")

    As I understand it, ∅ is a set containing no elements.
    Then I think it makes sense that |∅|=0. (Cardinality of empty set is zero).
    So... this is where things get hairy.

    1.
    What's the cardinality of {∅}? I think according to set theory that |{∅}|=1.
    This is because {∅} would contain itself right?

    2.
    If 1, then I'd assume that |{{∅}}|=2?

    Further, what would the cardinality of the following sets be:
    a. {∅,{∅}}
    b. {∅} ∩ {{∅}} (and what does this set contain? quite confusing)
    c. {{∅},∅} (this set is the same as a. I believe)

    Actually if anyone could also let me know which of these sets is equal I think it might help me figure it out. Anyway I'm just trying to undertand empty set a little because my textbook just glazes over it. Thanks~!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by hain
    Okay, I'm new to set theory and having a little trouble wrapping my head around how empty set works.

    (I'll represent empty set with the unicode character "∅")

    As I understand it, ∅ is a set containing no elements.
    Then I think it makes sense that |∅|=0. (Cardinality of empty set is zero).
    So... this is where things get hairy.

    1.
    What's the cardinality of {∅}? I think according to set theory that |{∅}|=1.
    Yes.

    This is because {∅} would contain itself right?
    No. {∅} contains ∅, {∅} and ∅ are not thesame set. The first is the
    set with one element which is the empty set, while the other is the empty
    set.

    2.
    If 1, then I'd assume that |{{∅}}|=2?
    No, {{∅}} has one element {∅}.
    Further, what would the cardinality of the following sets be:
    a. {∅,{∅}}
    |{∅,{∅}}|=2, {∅,{∅}} is a set with two elements ∅, and {∅}.
    b. {∅} ∩ {{∅}} (and what does this set contain? quite confusing)
    {∅} has one element ∅, {{∅} } has one element {∅}, and as ∅ is not
    equal to {∅} the intersection is empty so

    {∅} ∩ {{∅}} = ∅
    c. {{∅},∅} (this set is the same as a. I believe)
    Yes.

    Actually if anyone could also let me know which of these sets is equal I think it might help me figure it out. Anyway I'm just trying to undertand empty set a little because my textbook just glazes over it. Thanks~!
    RonL
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  3. #3
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    I see what your problem is. You are confused by the notion of an empty set. No one can answer that question. The "set" and "empty-set" are "undefined terms". Undefined terms are terms which cannot be rigorously defined. Rigorously defined means based on other mathematical definition. Mathemations need a place to start thus they chose the concept of the "set" as the place where everything else is defined. Note, it does not mean we cannot understand what a "set" is but rather we cannot "define" it properly. This, is perhaps what is giving you confusion.

    As an analogy, the terms "point" and "line" are undefined terms in geometry. This is perhaps why we have the 5 postulates of Euclid, not because we cannot understand them. But rather because we cannot prove them properbly because the condition for proving them requires us for having defined terms for "point" and "line".
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  4. #4
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    Thumbs up

    Hm, thanks.
    The theory is a bit confusing but I have a better idea of how sets work now
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