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Definition/Summary
Let A be a non-empty set. A collection P of non-empty subsets of A is called a partition of A if
1) For every S,T\in P we have S\cap T=\emptyset.
2) The union of all elements of P is A
We say a partition P is a collection because it is a "set of sets".
The elements of P are called the classes of the partition P.
Property two in the definition above says that each element of A is in at least one class of the partition P. Property one says that each element of A 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
P = \{ \{ x: x \text{ is a female student} \}, \{ y: y \text{ is a male student} \} \}
2) We can partition all the people of Earth according to the country they were born in.
3) We can partition the natural numbers \mathbb{N} into the even and odd numbers: P = \{ \{1,3,5,7,\ldots \}, \{ 2,4,6,8,\ldots \} \}; or we can partition the natural numbers according to whether they are prime or not: P = \{ \{2,3,5,7,11,13,\ldots \}, \{ 1,4,6,8,9,10\ldots \} \}
4) Although there is often some property common to the elements of a certain class, this is not necessary. For example, the set A = \{ 12.3, -1, \pi, apple, 7^{8/3}, 912312 \} can be partitioned as such: P = \{ \{ 12.3, apple, 912312 \}, \{ -1, \pi, 7^{8/3} \} \}
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!
Let A be a non-empty set. A collection P of non-empty subsets of A is called a partition of A if
1) For every S,T\in P we have S\cap T=\emptyset.
2) The union of all elements of P is A
We say a partition P is a collection because it is a "set of sets".
The elements of P are called the classes of the partition P.
Property two in the definition above says that each element of A is in at least one class of the partition P. Property one says that each element of A 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
P = \{ \{ x: x \text{ is a female student} \}, \{ y: y \text{ is a male student} \} \}
2) We can partition all the people of Earth according to the country they were born in.
3) We can partition the natural numbers \mathbb{N} into the even and odd numbers: P = \{ \{1,3,5,7,\ldots \}, \{ 2,4,6,8,\ldots \} \}; or we can partition the natural numbers according to whether they are prime or not: P = \{ \{2,3,5,7,11,13,\ldots \}, \{ 1,4,6,8,9,10\ldots \} \}
4) Although there is often some property common to the elements of a certain class, this is not necessary. For example, the set A = \{ 12.3, -1, \pi, apple, 7^{8/3}, 912312 \} can be partitioned as such: P = \{ \{ 12.3, apple, 912312 \}, \{ -1, \pi, 7^{8/3} \} \}
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!