Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lectures we described "bounded" and "unbounded" intervals.
Bounded intervals are limited by their endpoints to a finite length on the number line.
Bounded intervals can be closed
open
left-closed right-open
or left-open right-closed.
Unbounded intervals can be right-unbounded and left-closed or open
left-unbounded and right-closed or open
or unbounded on both ends.
These intervals are called "proper intervals" because they include more than one point.
However, not all intervals are proper.
It is possible for an interval to include only one point, or no points.
If the left and right endpoints of a closed interval are the same number
then the interval is called a "degenerate interval"
and represents a single point on the number line.
So a degenerate interval is a set which contains one real number.
On the other hand, if both endpoints of an open interval are the same
then that interval contains no points so this interval represents the empty set.
When we first introduced interval notation
we said that an interval was a special type of set
consisting of a continuous span of numbers on the number line.
Mathematicians call this property "connectedness".
Intervals are the connected sets of the number line.
Even though "a continuous span" is intuitively easy to understand
a mathematician might define the concept of an interval in the following way:
"An interval is a set of real numbers
with the property that any number that lies between two numbers in the set
is also included in the set."
So the key property of an interval is that
no matter which two numbers we pick in the interval
every number between those two numbers must also be contained in the interval.
Since the two numbers we pick can be as close as we like to the endpoints
no numbers which lie between the endpoints can be excluded from the interval.
Therefore, every number between the two endpoints of an interval
must be included in the interval.
This property is true whether the endpoints are included in the interval
or not included.
For unbounded intervals, since one can pick two numbers arbitrarily far apart
all real numbers must be included in the interval.
This definition is useful for testing if a set of real numbers
can be represented by a single interval.
For example, let's say that we have a set which consists of zero and two
and all numbers in between except the number one.
Using set-builder notation this would be the set of all numbers, x
such that x is greater than or equal to zero and less than or equal to two
and not equal to one.
But this set is not an interval.
We know this because our definition for intervals says that
every number that lies between any two numbers in the set must also be included in the set.
However, the number one lies between numbers in this set, but is not included in the set.
So this set cannot be described by a single interval.
However, sometimes it is easier and more clear
to describe regions of the number line like this set, using interval notation.
This can be accomplished by combining intervals.
Since intervals are sets, we can use set operations to combine multiple intervals into one set.
For example, we can take the left-closed right open interval between zero and one
and the left-open right-closed interval between one and two
and take their union to create the desired set.
As another example, let's create one set containing all the numbers between zero and one
and between two and three but not including the endpoints.
In set-builder notation this is the set of all numbers, x
such that x is greater than zero and less than one
or greater than two and less than three.
If we pick one number between zero and one and a second number between two and three
any number between one and two lies between these two numbers
but is not a member of the set.
Therefore, this set cannot be represented by a single interval.
However, if we take the open interval between zero and one
and the open interval between two and three
then the union of these two intervals creates the desired set.
As a final example, let's create a set containing all real numbers except zero.
In set-builder notation this is the set containing all numbers x, not equal to zero.
This set is not an interval, since if we pick any two numbers on either side of zero
both numbers are members of the set
but the number zero, which lies between these two numbers, is not a member of the set.
We can represent this set using interval notation
by taking the union of the left-unbounded interval
containing all numbers less than zero
and the right-unbounded interval containing all numbers greater than zero.
Intervals which are neither empty nor degenerate are said to be "proper".
As we have seen, all proper intervals contain more than one point.
If an interval contains more than one point
it must contain an infinite number of points
since between any two distinct points on the number line
there are an infinite number of points.
Therefore all proper intervals are sets containing an infinite number of elements.
So far we have talked about sets of real numbers
and we have seen that these sets can be visualized using the number line.
A line is a one-dimensional object
and the number line represents real numbers as points in this one-dimensional space.
In the next lecture, we will examine mathematical objects
which can be visualized as points in two or more dimensions.