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:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)