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What is a partition

  1. Jul 23, 2014 #1
    Definition/Summary

    Let [itex]A[/itex] be a non-empty set. A collection [itex]P[/itex] of non-empty subsets of [itex]A[/itex] is called a partition of [itex]A[/itex] if
    1) For every [itex]S,T\in P[/itex] we have [itex]S\cap T=\emptyset[/itex].
    2) The union of all elements of [itex]P[/itex] is [itex]A[/itex]

    We say a partition [itex]P[/itex] is a collection because it is a "set of sets".

    The elements of [itex]P[/itex] are called the classes of the partition [itex]P[/itex].

    Property two in the definition above says that each element of [itex]A[/itex] is in at least one class of the partition [itex]P[/itex]. Property one says that each element of [itex]A[/itex] is in exactly one class of the partition.

    Equations



    Extended explanation

    A partition of a set is breaking down of the set into distinct parts.

    Examples:

    1) Let A be the set of all the students in a lecture. We can partition A into two classes; one class of all the females, and the other class comprised of all the males. The partition of A is then
    [itex]P = \{ \{ x: x \text{ is a female student} \}[/itex], [itex]\{ y: y \text{ is a male student} \} \} [/itex]

    2) We can partition all the people of Earth according to the country they were born in.

    3) We can partition the natural numbers [itex]\mathbb{N}[/itex] into the even and odd numbers: [itex]P = \{ \{1,3,5,7,\ldots \}, \{ 2,4,6,8,\ldots \} \}[/itex]; or we can partition the natural numbers according to whether they are prime or not: [itex]P = \{ \{2,3,5,7,11,13,\ldots \}, \{ 1,4,6,8,9,10\ldots \} \}[/itex]

    4) Although there is often some property common to the elements of a certain class, this is not necessary. For example, the set [itex]A = \{ 12.3, -1, \pi, apple, 7^{8/3}, 912312 \}[/itex] can be partitioned as such: [itex]P = \{ \{ 12.3, apple, 912312 \}, \{ -1, \pi, 7^{8/3} \} \}[/itex]

    See also the entry on equivalence relations, to which partitions are closely related.
    Specifically, the quotient set of an equivalence relation is a partition of the underlying set; conversely, a partition of a set defines an equivalence relation on that set, two elements being in relation to one another if and only if they belong to the same class of the partition.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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