Disjoint Sets and Two More Operations
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 6.1. 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 that 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 
, is the set of all those elements in the universal set that are not in A.
Example 2
If the universal set is the set {1, 2, 3, …, 10} and A = {1, 3, 5, 7, 9}, then
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.
Operation 4: Subtraction of Sets
An operation between two sets that is not in your textbook is subtraction.
Subtraction – The set A - B (read "A minus B") is the set of all elements in A that are not in B.
Example 3
If A = {1, 2, 3, 4} and B = {1, 3, 5, 7, 9}, then A – B = {2, 4}.
Note: All we have left in the new set A – B are the elements in A that are not in B. Most people have little difficulty taking away the 1 and the 3 in this case, but sometimes wonder if they should be doing something with the 5, 7, and 9 from set B. If you remember that -- when you subtract -- you do not add anything, it will be a little easier to do.
Here is a little story I tell my classes to help them remember how to do subtraction of sets.
There is a campus party going on and several mothers find out about it and go to the party to take their daughters home. Using our example above, let set A represent the girls who were at the party and set B be the mothers who arrived to get their daughters. When the mothers arrive at the party those who had daughters there took them home, while those who did not have daughters at the party went home alone. In our example, girls 2 and 4 did not have their mothers come for them so they stayed at the party. Mothers 5, 7, and 9, even though they may have been tempted, did not stay at the party but went home, leaving only girls 2 and 4.
This story may sound a little silly to you, but if you remember it you will have very little trouble doing subtraction of sets.
Let's try a few more examples.
Example 4
If A = {a, b, c} and B = {b, c, d, e}, then A – B = {a} but B - A = {d, e}.
Example 5
Let A = {2, 5, 7, 9}, B = {1, 3, 5, 7, 8}, and C = {1, 2, 3, 4, 5, 6}.
Then, (A∪B ) – (B∩C) = {2, 7, 8, 9}
{1, 2, 3, 5, 7, 8, 9} – {1, 3, 5} = {2, 7, 8, 9}
(A∩C) – (B∪C)={ }
{2, 5} – {1, 2, 3, 4, 5, 6, 7, 8} = Ø.
Example 6
What is the result of subtracting { } from a non-empty set D? The answer should be fairly obvious when you subtract nothing from something you get the original something so D – { } = D. In a similar sense { } – D should be { }. Remember: When you subtract D from { } you cannot add anything to { } from D; you can only take the common elements away.