Hello. I'm Professor Von Schmohawk and welcome to Why U. So far we have seen how to define a set either by listing its members or by using set-builder notation to state the properties that every element of the set must satisfy. We also described several types of relations that can exist between sets.
In this lecture we will describe several operations which can be performed on sets such as intersections and unions and see how these relations and operations can be visualized graphically by using what is called a Venn diagram. In a Venn diagram, sets are usually represented by circles or other types of enclosed areas. The interior of the circle represents all the elements of the set while the exterior represents any elements that are not members of the set. For instance, let's say that we have two sets, A and B.
Drawing the sets so that they don't overlap implies that the two sets contain completely different elements. In this case, we say that the two sets are disjoint. Disjoint sets have no elements in common. If the two circles overlap this indicates that there are one or more elements common to both sets.
The intersecting area represents elements that are members of both A and B. This collection of common elements is called the intersection of sets A and B. We denote set intersection by a symbol which looks like an inverted U.
This collection of common elements forms a new set. For example, if set A contains the elements 1, 2, and 3 and set B contains 2, 3, and 4 then the elements which A and B have in common are 2 and 3 and this intersection forms a new set containing the elements 2 and 3. Two disjoint sets have no elements in common so their intersection forms the empty set. Forming the intersection of two sets is a binary set operation.
in the same way that forming the sum of two numbers is a binary numerical operation. The result of this set operation is another set just as the result of this numerical operation is another number. Using set-builder notation, we can write the definition of an intersection of two sets formally as the intersection of sets A and B is the set of all elements x such that x is a member of A and x is a member of B.
Another important set operation is the union of sets. The union of two sets, A and B, includes all elements which are members of either A or B. If sets A and B have any elements in common then these elements which are members of but both sets are only included once in the union. For example, if set A contains the elements 1, 2, and 3 and set B contains 2, 3, and 4 then the elements which are members of A or B are 1, 2, 3, and 4. This union forms a new set containing the elements 1, 2, 3, and 4. Notice that even though 2 and 3 are members of of both sets A and B, two and three are only listed once when we write the union. Using set builder notation, we can write the definition of the union of two sets formally as the union of sets A and B is the set of all elements x such that x is a member of A or x is a member of B.
Subsets and supersets can can also be represented using Venn diagrams. In a Venn diagram, a subset is typically shown as a smaller region within the larger superset. For instance, in this diagram, set A is a subset of B and set B is a superset of A.
If we use a Venn diagram to illustrate the relations between the sets of numbers we have studied in pre-algebra, it might look like this. Remember that in a Venn diagram although sets are usually represented by circles or ovals they can be illustrated by any type of enclosed area. This diagram shows that the set of natural numbers is a subset of the set of whole numbers which is a subset of the set of rational numbers. The union of the set of rational numbers and the set of irrational numbers forms the set of real numbers. So far we have seen how various relations between sets as well as set operations such as intersections and unions can be represented using Venn diagrams.
In the next lecture we will see how Venn diagrams can be used to visualize more complex operations.