Set theory question

Discussion in 'Physics & Math' started by Rick, Aug 25, 2009.

  1. Rick Valued Senior Member

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    Hi,

    I am trying to read up on set theory and I stumbled upon a problem (it seems simple enough) but still:

    A UNION B = B UNION A

    I am looking for a generic method to solve this, like if say we have:

    A INTERSECT (B UNION C) = (A INTERSECT B) UNION (A INTERSECT C)

    Suggestions?
     
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  3. funkstar ratsknuf Valued Senior Member

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    In naïve set theory those equalities are easy to prove: Expand the definitions of union and intersection, and use the properties of the logical connectives.

    E.g., \(A \cup B \stackrel{(1)}{=} \{ x \mid x \in A \wedge x\in B \} \stackrel{(2)}{=} \{ x \mid x \in B \wedge x\in A \} \stackrel{(1)}{=} B \cup A\), where (1) is by the definition of union, and (2) uses that logical conjunction is commutative.

    In axiomatic set theory (such as NBG or ZF(C)) things are a little more hairy, but I'll assume that you didn't mean those, since you didn't mention them.
     
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  5. Rick Valued Senior Member

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    hmmph understood (I haven't reached ZF yet

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    ), some more questions:

    What is an empty set? if U is a universal set, which is the set of all sets, then empty set has to belong to Universal set, that DOESNT MAKE SENSE TO ME!

    How can a set without any elements belong to a set of all elements?! Can someone explain:
    1.) What is an empty set
    2.) What is X U {}
    3.) What is X Intersect { }
    4.) What is relationship between Universal set and an empty set?

    Sorry if my questions are a little dumb.

    Rick
     
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  7. noodler Banned Banned

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    The empty set is a container which is empty. But it contains the empty set.
    The universal set also contains the above set, or intersects with it. Intersection is disjoint union - the empty set is in the universal set, but stays the empty set.

    The intersection is "empty", you see.
     
  8. Rick Valued Senior Member

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    3,336
    well it still bothers me, because how can a set be called a set if its empty, I thought the definition of set was collection of things / elements that obey a certain "rule" to map to resulting set of elements. (rule = function, I am not even going to think about functions now ... its a whole new ball game for me)?

    OK something that follows up is, so all sets belong to Universal set,then can we infer that Universal set should belong to itself?!!

    Rick
     
  9. noodler Banned Banned

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    751
    A set can be called empty if it has no members. The two sets are like this: the universal set has all members in it which are not in the empty set.

    Therefore all things not in the universal set are in the empty set; since a thing is in the unversal set by definition, the null-set or nullspace is always empty = {}.
    The {} is equivalent to 0, or {0}. You can think of a 0 as a container with nothing in it - a boundary or maybe an empty pipe - there is a container but nothing can go in it because, if it's a physical thing it's already in the U set.

    And yes, all sets have a subset which is also the set; all sets are "themselves" and include themselves as a member. A way to think of this is: a set is a copy of itself.
     
  10. temur man of no words Registered Senior Member

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    1,330
    You can think of the empty set as conditions which nothing can satisfy, like, set of all x such that x>1 and x<0 etc. You can do without empty set but then there will be many exceptions and special cases in the theorems.

    Way to go! I urge you to think a bit harder; you will discover something very interesting.
     
  11. funkstar ratsknuf Valued Senior Member

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    That's not true. The empty set is not an element of itself, but it does have itself as a subset.
     
  12. Rick Valued Senior Member

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    3,336
    Temur,

    I am going to delve deeper into it, thats the fun part of "independent" study, you can delve deep into a concept and hold onto it till you not only get it, but perhaps develop some philosophical ideas about it.

    Very interesting. I'll ask questions as a follow up, cuz I am a noob anyways.

    Rick
     
  13. noodler Banned Banned

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    751
    Ok, then a subset of the empty set, is the empty set; therefore the set is itself = includes itself which is what I meant;

    How many subsets are there? An infinite number of empty sets are in there therefore it contains (an infinite #) of itself. The empty set contains itself as a subset = the empty set contains itself by remaining empty, up to infinity. All subsets are the set = the set's elements are all {}. So it contains {} (emptily).

    I always think about how many empty boxes, have empty boxes in them. Is it the box which is empty or the space in the box? How many empty spaces are there in an empty space?

    So I should have posted: "the empty set contains itself as a subset, the set and all subsets of it are infinitely identical"
     
    Last edited: Aug 27, 2009

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