Two Basic Operations on Sets

In arithmetic, we have the binary operations of addition, subtraction, multiplication, and division; in set theory we also have operations. Two basic operations are called union of sets and intersection of sets.

Operation 1: Union of Sets

The union of two sets A and B results in another set that includes all of the elements in A together with all of the elements in B. In other words all of the elements that are in A or in B. The symbol we use for union of sets is ∪.

Example 1

Suppose that A = {1, 2, 3} and B = {1, 3, 5, 7} then A∪B would be the set {1, 2, 3, 5, 7}.

Does it make sense to say that A∪{ } = A? If you take all of A and add nothing, what do you get? The set A, of course.

Operation 2: Intersection of Sets

The second basic operation on two sets is intersection. By the intersection of sets A and B we mean the set having those elements that are common to both A and B. The symbol we use for intersection is ∩.

Example 2

If we have a set A = {1, 3, 5, 7} and B = {3, 4, 5}, then A∩B would be the set {3, 5}. Note, these are the only elements common to both sets.

Finding the intersection of sets is like sorting socks after doing a wash. You have two piles of socks from the wash and you put the matching socks together and set them in the laundry basket to distribute to the family. The paired socks in the laundry basket represent the intersection of the original two piles of socks.

Earlier, we talked about the union of a set A and the empty set resulting in set A. Can you see what the result would be if we intersect a set A with the empty set? If you think for a minute, it should be obvious: the intersection must be the empty set, since you cannot have anything in common with a set containing no elements.