n. 1a. a set of disjoint and exhaustive subclasses of a given class that divide it in such as way that each member of the given class is a member of exactly one such subclass. Such a division is possible if and only if there is an equivalence relation that relates two elements of the underlying class if and only if they are members of the same subclass. For example, since two integers are congruent modulo n if and only if their difference is divisible by n, congruence mod n is reflexive (a≡a for all a), transitive (if a≡b and b≡c, then a≡c), and symmetric (a≡b if and only if b≡a), so that this is an equivalence relation; hence it can be proved that, since every integer has a unique smallest positive remainder when divided by n, and so is congruent mod n to only one of the integers between 0 and n - 1, these congruence classes are both disjoint and exhaustive of all the integers, and so constitute a partition of them. See also covering.
b. such a division of a class into a set of subsets. 2. a division of a given positive integer into positive integral parts. For example,
5=4+1=3+2 =3+1+1= 2+2+1=2+1+1+1=1+1+1+1+1
exhibits the seven possible partitions of 5. See also partition function. 3. a division of a given matrix into conformable submatrices. 4. a finite sequence of points {x_k } of a given interval [a, b] such that A partition of an interval therefore yields a finite number of subintervals that are pairwise disjoint.