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.
