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Why the elements of a set have to be distinct?

  1. Jul 7, 2012 #1

    I haven't taken any set theory course during my studies and this is a query of mine, though to many of you that are familiar with this subject might seem naive.

    Any suggestions would be useful.

    Many thanks
  2. jcsd
  3. Jul 7, 2012 #2


    Staff: Mentor

    Think of it this way a class room of thirty students as a set of thirty students vs the set of student first names. The members of the student set may have the same first names so the first names set will have less elements.

    If someone asked you how many unique first names there were in the classroom then you could simply count the elements in the set. Hence a set is defined to have unique elements for queries such as these.

    Here's another reference to read from dr math

  4. Jul 7, 2012 #3
    Because we defined it that way; because it's useful. A set is a mathematical construct that tags every object in the Universe with either "member" or "not a member". Its purpose isn't to deal with quantities of specific object; we have other concepts to deal with that.
  5. Jul 8, 2012 #4
    It is a consequence of the Pairing and Extensionality Axioms, for instance:
    [tex] \{ x, y, z \} = \{ x,y,y,z \}= \{ x,y,y,y,z \}= \{ x,\mbox{ ... any number of y's ... },z \} [/tex]
  6. Jul 8, 2012 #5
    Here's a thought that might help you out. Imagine you created a set i^4n for all n in N. Since there are an infinite number of n, there are an infinite number of elements in that set, each distinct. But if you "physicalized" those elements they would all look exactly like the number 1. To the naked eye they would seem indistinguishable, even though they would all be distinct in the sense, for instance, that each element of the set could be interpreted as representing one rotation around a circle. So... it kind of matters the sense in which one is using the term "indistinguishable."

    Mathematically, anyway, there needs to be some parameter that differentiates each element in a set.

    - AC
  7. Jul 8, 2012 #6
    The axioms far post-date the concept of a set. Even naive set theory doesn't keep track of element multiplicity.
  8. Jul 8, 2012 #7
    You're right, but notationally, the elements need not be distinct. The definition of set makes them distinct, regardless of notation.

    The question posed is "why do the elements of a set have to be distinct?"

    The answer is "You are asking the wrong question."

    He will come across notational conveniences that imply indistinctness without realizing that set membership is independent of the notion of distinction.
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