In mathematics, the partition of a line is a sorting of its elements into non-empty subsets, in such(a) a way that every element is pointed in precisely one subset.
Every equivalence relation on a set defines a partition of this set, as alive as every partition defines an equivalence relation. A manner equipped with an equivalence version or a partition is sometimes called a setoid, typically in type theory & proof theory.
A partition of a set X is a set of non-empty subsets of X such(a) that every element x in X is in exactly one of these subsets i.e., X is a disjoint union of the subsets.
Equivalently, a family of sets P is a partition of X whether and only if any of the following conditions hold:
The sets in P are called the blocks, parts, or cells, of the partition. if then we exist the cell containing a by . That is to say, is notation for the cell in P which contains a.
Every partition, P, may be subject with an equivalence representation on X, namely the relation such that for all we create if and only if equivalently, if and only if . The notation evokes the image that the equivalence relation may be constructed from the partition. Conversely every equivalence relation may be identified with a partition. This is why this is the sometimes said informally that "an equivalence relation is the same as a partition". If P is the partition identified with a condition equivalence relation , then some authors write . This notation is suggestive of the conviction that the partition is the set X divided up up in to cells. The notation also evokes the idea that, from the equivalence relation one may clear the partition.
The rank of P is | − ||, if X is finite.
then we exist the cell containing a by ![[a]](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea82bc70a8e322f13a3c4e5b9d5d69e8ef097ad8)
. That is to say, 
such that for all 
we create 
if and only if ![{\displaystyle a\in [b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8abb5047c33633d5a0697408f3df2aead151e4ea)
equivalently, if and only if ![{\displaystyle b\in [a]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1c441d4221216c8947ac50d43331042f72672b)
. The notation 
, then some authors write 
. This notation is suggestive of the conviction that the partition is the set X divided up up in to cells. The notation also evokes the idea that, from the equivalence relation one may clear the partition.