Disjoint Sets and Complementary Sets

Before we move on to two more operations, let's learn what "disjoint sets" are.

Disjoint Sets

Two sets A and B are called disjoint if they have no elements in common.

In this case the intersection will be the empty set. Now look again at Figure 1 in section 7.1. Are sets A and B disjoint? This is more difficult to answer because they are showing an area of intersection for the two sets in the Venn diagram. Since this particular Venn diagram is only showing the possibility that sets A and B could have some elements in common, we cannot say for sure whether A and B are disjoint or not.

Example 1

If we knew that set A contained the elements {1, 2, 3} and set B contained the elements {5, 6, 7}, then the intersection of sets A and B would be { }, which is the empty set. So, in this case sets A and B are disjoint.

 

Operation 3: Complement of a Set

The complement of a set A, written $\overline{A}$ , is the set of all those elements in the universal set, U, that are not in A.

Example 2

If the universal set is the set {1, 2, 3, …, 10} and $A=\left\{1,3,5,7,9\right\}$ , then  $\overline{A}\text{= {2, 4, 6, 8, 10}}$ .

If you think about it, it is apparent that any set and its complement will always be disjoint.

What is the complement of the universal set? The empty set.