Partition of A Set - Examples

Examples

• Every singleton set {x} has exactly one partition, namely { {x} }.
• For any nonempty set X, P = {X} is a partition of X, called the trivial partition.
• For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, namely, {A, U−A}.
• The set { 1, 2, 3 } has these five partitions:
  • { {1}, {2}, {3} }, sometimes written 1|2|3.
  • { {1, 2}, {3} }, or 12|3.
  • { {1, 3}, {2} }, or 13|2.
  • { {1}, {2, 3} }, or 1|23.
  • { {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number).
• The following are not partitions of { 1, 2, 3 }:
  • { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the empty set.
  • { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block.
  • { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partition of {1, 2}.