What Is the Intersection of Two Sets?

When dealing with set theory, there are a number of operations to make new sets out of old ones. One of the most common set operations is called the intersection. Simply stated, the intersection of two sets A and B is the set of all elements that both A and B have in common.

We will look at details concerning the intersection in set theory.  As we will see, the key word here is the word "and."

An Example

For an example of how the intersection of two sets forms a new set, let’s consider the sets A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7, 8}. To find the intersection of these two sets, we need to find out what elements they have in common. The numbers 3, 4, 5 are elements of both sets, therefore the intersections of A and B is {3. 4. 5].

Notation for Intersection

In addition to understanding the concepts concerning set theory operations, it is important to be able to read symbols used to denote these operations. The symbol for intersection is sometimes replaced by the word “and” between two sets. This word suggests the more compact notation for an intersection that is typically used.

The symbol used for the intersection of the two sets A and B is given by A ∩ B. One way to remember that this symbol ∩ refers to intersection is to notice its resemblance to a capital A, which is short for the word "and."

To see this notation in action, refer back the above example. Here we had the sets A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7, 8}. So we would write the set equation A ∩ B = {3, 4, 5}.

Intersection With the Empty Set

One basic identity that involves the intersection shows us what happens when we take the intersection of any set with the empty set, denoted by #8709. The empty set is the set with no elements. If there are no elements in at least one of the sets we are trying to find the intersection of, then the two sets have no elements in common. In other words, the intersection of any set with the empty set will give us the empty set.

This identity becomes even more compact with the use of our notation. We have the identity: A ∩ ∅ = ∅.

Intersection With the Universal Set

For the other extreme, what happens when we examine the intersection of a set with the universal set? Similar to how the word universe is used in astronomy to mean everything, the universal set contains every element. It follows that every element of our set is also an element of the universal set. Thus the intersection of any set with the universal set is the set that we started with.

Again our notation comes to the rescue to express this identity more succinctly. For any set A and the universal set U, A ∩ U = A.

Other Identities Involving the Intersection

There are many more set equations that involve the use of the intersection operation. Of course, it's always good to practice using the language of set theory. For all sets A, and B and D we have:

• Reflexive Property: A ∩ A =A
• Commutative Property: A ∩ B = B ∩ A
• Associative Property: (A ∩ B) ∩ D =A ∩ (B ∩ D)
• Distributive Property: (A ∪ B) ∩ D = (A ∩ D)∪ (B ∩ D)
• DeMorgan’s Law I: (A ∩ B)^C = A^C ∪ B^C
• DeMorgan’s Law II: (A ∪ B)^C = A^C ∩ B^C