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).

Two or more sets are mutually disjoint if any two of the sets in question are disjoint. For example, {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 number that belongs to all of them.

We can 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.