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What is an indexed family of sets. I need a simple example

  1. Dec 12, 2011 #1
    I have looked it in the Wikipedia, but no simple example. So I am not sure. Is the indexed family of sets just power sets, indexed (indexing means labeling as I understand)?

    For example the indexed family of sets of set A ={1,2,3,4,5,6} is just the collection of element from power set. A sub 1 may be {1} and A sub 7 may be {1,2} and so on. Indexed family of sets may be the collection of those sets as I understand. Can anyone clarify this please.(I am not a math major.)
     
  2. jcsd
  3. Dec 12, 2011 #2

    LCKurtz

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    Here's an example that may help. For r > 0 define[tex]
    A_r = \{(x,y):x^2 + y^2 < r^2\}[/tex]This gives an uncountable family of nested discs indexed by their radius.
     
  4. Dec 12, 2011 #3
    Let the group of sets be called G. In G, there are five sets, G1, G2, G3, G4, and G5. Let those sets be the following:
    G1: {2, 4, 6, 8,}
    G2: {3, 6, 9, 12}
    G3: {4, 8, 12, 16}
    G4: {5, 10, 15, 20}
    G5: {6, 12, 18, 24}

    So, G: {G1, G2, G3, G4, G5}

    This is a family of sets. I think the index refers to the sub number. In paper/pencil land for G1, the 1 would be a subscript.
     
  5. Dec 12, 2011 #4

    HallsofIvy

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    On this board, you an do it with the html code C[ sub]1[ /sub ] without the spaces: C1. Or do it using the tex code: [ tex ]C_1[ /tex ] without the spaces gives [itex]C_1[/itex].
     
  6. Dec 12, 2011 #5
    {Alice, Bob, Carla} share a house, but they're not always all in. The set of possible occupant sets of the house can be indexed [0..7] with A+2B+4C.
     
  7. Dec 13, 2011 #6
    Is it just the family of set with index notation? Wiki gave me "Let S be a set. An indexed family of sets {Ci}iεI is an indexed family that maps I to elements of the power set of S.

    Hence, an indexed family of sets is conceptually different from a family of sets (which is just a synonym for "set of sets"), but in practice the distinction is sometimes fuzzy and the indexed family is identified with its range and treated like an ordinary family."
     
  8. Dec 14, 2011 #7
    Ah, I messed up. I will now use my text book to define it.

    "Let [itex]\Delta[/itex] be a non-empty set such that for each [itex]\alpha[/itex] [itex]\in[/itex] [itex]\Delta[/itex] there is a corresponding set A[itex]\alpha[/itex]. The family {A[itex]\alpha[/itex]: [itex]\alpha[/itex] [itex]\in[/itex] [itex]\Delta[/itex]} is an indexed family of sets. The set [itex]\Delta[/itex] is called the indexing set and each [itex]\alpha[/itex] [itex]\in[/itex] [itex]\Delta[/itex] is an index."

    So my delta was {1, 2, 3, 4, 5}. That is, my indexing set was {1, 2, 3, 4, 5}. It could have easily been all natural numbers or some other known set, and then I could say my indexing set was N, that is, Natural Numbers. My family is all Gi, such that i is an element of {1, 2, 3, 4, 5}, that is, my family is: {Gi: i [itex] \in [/itex][itex]\Delta[/itex]} (if I want my set {1, 2, 3, 4, 5} to be named delta. it doesn't have to be named that.) . Thus, my family is {G1, G2, G3, G4 G5}. The numbers 1, 2, 3, 4, and 5 are the indices. Each is an index.
     
  9. Dec 15, 2011 #8
    Thanks ArcanaNoir for the response. I also like LCKurtz' example where Δ is all positive real number.
     
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