Answer to Question #289900 in Real Analysis for scholar

Question

Define partition

Answer 1

A partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.

Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. 

For example:

• 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 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