Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far, we have seen how set builder notation can be used to define sets.
For example, we could define a set as follows:
Set A is the set of all elements, x
such that
x is a real number
greater than or equal to zero
and less than or equal to three.
We also saw how Venn diagrams can be used to illustrate relations between sets.
This Venn diagram shows that set A is a subset of the set of real numbers.
However, it does not show anything about the elements of A
having values greater than or equal to zero and less than or equal to three.
Venn Diagrams are a good way to visualize relations and operations
for sets whose elements can be any type of object.
However, in Algebra, the sets of primary interest are sets of numbers.
Number lines are a good way to visualize these numerical sets.
The number line represents the set of all real numbers.
Every real number has a corresponding point on the number line.
As an example, let's use the number line to illustrate set A.
As we saw in the previous lecture
in cases where the universal set is obvious from the context
it may be omitted from the set definition.
In this case, the universal set is the set of all real numbers.
Since it is understood that all numbers illustrated by the number line are real
we can omit this from our set definition.
Zero and three are the least and greatest numbers in set A
so we place solid dots at those points on the number line.
Since set A also contains all real numbers between zero and three
we draw a solid line between the two endpoints.
This set of numbers forms what is called an "interval".
An interval is a set whose members are a continuous span of numbers on the number line.
"Finite intervals" are bounded by two endpoints.
In this example, the endpoints are zero and three.
The endpoints of an interval don't necessarily have to be included in the interval.
However, since set A contains numbers greater than or equal to zero
and less than or equal to three
the endpoints zero and three are included in this interval.
Intervals which include their endpoints are called "closed intervals".
Intervals which do not include their endpoints are called "open intervals".
The endpoints of open intervals are drawn as open circles
indicating that they are not members of the set.
This interval corresponds to the set of real numbers greater than zero and less than three.
Intervals can also have one included endpoint and one excluded endpoint or vise versa.
It is not always necessary to use set-builder notation when defining numeric sets.
Another notation, called "interval notation"
is often an easier and more compact way to define numeric sets.
As an example, let's use interval notation
to define a closed interval whose endpoints are zero and three.
Using interval notation we simply write the two endpoints
separated by a comma and enclosed in square brackets.
Open intervals are written using parentheses.
And if the interval is closed on one end and open on the other
then the closed end with the included endpoint is written with a bracket
and the open end with the excluded endpoint is written with parentheses.
So far we have seen how to write four different types of intervals.
Let's use a and b to represent the left and right endpoints.
If both endpoints are excluded then the interval is "open".
Since the endpoints a and b are excluded
the set consists of all numbers greater than a
and less than b.
If both endpoints are included then the interval is "closed".
Since the endpoints a and b are included
the set consists of all numbers greater than or equal to a
and less than or equal to b.
If the interval includes the left endpoint but not the right
then we say that the interval is "left-closed and right-open".
Since a is included and b is excluded
the set consists of all numbers greater than or equal to a
and less than b.
And if the interval includes the right endpoint but not the left
then the interval is "right-closed and left-open".
Since a is excluded and b is included
the set consists of all numbers greater than a
and less than or equal to b.
We call all these intervals "finite" or "bounded"
since each interval has both an upper and lower bound
which limit the interval to a finite length on the number line.
In the next lecture we will explore the properties of "unbounded" intervals.