Definition
A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets.
Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and:
- The union of the elements of P is equal to X. (The elements of P are said to cover X.)
- The intersection of any two distinct elements of P is empty. (We say the elements of P are pairwise disjoint.)
In mathematical notation, these two conditions can be represented as
- 1.
- 2.
where is the empty set.
The elements of P are called the blocks, parts or cells of the partition.
The rank of P is |X| − |P|, if X is finite.
Read more about this topic: Partition Of A Set
Famous quotes containing the word definition:
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)