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 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 both sets A and B
2 and 3 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 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.