Partition (mathematics)

In mathematics, partition refers to two related concepts, in set theory and number theory.

Partition (set theory)
A partition of a set X is a collection $$\mathcal{P}$$ of subsets of X such that every element of X is in exactly one of the subsets in $$\mathcal{P}$$.

Hence a three-element set {a,b,c} has 5 partitions:


 * {a,b,c}
 * {a,b}, {c}
 * {a,c}, {b}
 * {b,c}, {a}
 * {a}, {b}, {c}

Partition (number theory)
A partition of an integer n is an expression of n as a sum of positive integers, with the order of the terms in the sum being disregarded.

Hence the number 3 has 3 partitions:


 * 3
 * 2+1
 * 1+1+1