Partition Definition: What is a Partition?

[Greg Bernhardt]

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.

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[fresh_42]

A related and important concept (in differential geometry) is the partition of unity.