Disjoint sets

In mathematics, two sets are said to be disjoint if they have no element in common. For example, {1,2,3} and {4,5,6} are disjoint sets.

The following statements are logically equivalent:

• A and B are disjoint.
• The intersection of A and B is the empty set.
• A ∩ B = {} (the same as the above, but in symbols).

Given several sets, we say they are mutually disjoint or pairwise disjoint if any two of the sets in question are disjoint. For example, the sets {1,2,3}, {4,5,6}, and {7,8,9} are mutually disjoint. However, {1,2,3}, {4,5,6}, and {3,4} are not mutually disjoint, even though there is no element that belongs to all of them.

We also say that a set U whose elements are themselves sets is mutually disjoint if its members are mutually disjoint. In symbols:

For any A,B in U, A = B or A ∩ B = {}.

U is a partition of a set X if:

• the union of U is X;
• U is mutually disjoint (as above); and
• {} does not belong to U.