Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lecture we saw how to create bounded intervals
using either set-builder notation or interval notation.
We saw that if an interval includes both its endpoints, the interval is said to be "closed".
In interval notation, included endpoints are enclosed in square brackets.
Intervals which do not included their endpoints are said to be "open".
Excluded endpoints are enclosed in parentheses.
It is also possible for one endpoint to be included and the other excluded.
If the left endpoint is included or excluded
the interval is said to be "left-closed" or "left-open"
and if the right endpoint is included or excluded
the interval is "right-closed" or "right-open".
Bounded intervals are also called "finite intervals"
since their endpoints limit them to a finite length on the number line.
Endpoints create bounds for an interval
in the same way that a fence creates a boundary for a plot of land.
Just as an interval may or may not include its endpoints
a plot of land may or may not include a fence which creates its boundary.
However, whether or not a plot of land includes the fence which creates it's boundary
it is still bounded by the fence and therefore limited to a finite area.
Likewise, whether or not an interval includes its endpoints
those endpoints bound the interval to a finite length on the number line.
So left and right endpoints act as lower and upper bounds for the numbers in an interval.
No number contained in an interval is less than its lower bound
or greater than its upper bound.
In a closed interval the left endpoint is the "minimum"
or smallest number in the interval
and the right endpoint is the "maximum" or largest number in the interval.
However since an open interval does not contain its endpoints
open intervals have no minimum or maximum element.
To see why, let's take a closer look at an open interval.
This interval's right endpoint is the number one.
However, since one is not included in the interval
in order to find the maximum number in this interval
we must find the largest number less than one.
If we pick zero-point-nine as the maximum number
we know that this is incorrect since zero-point-nine-nine is larger
but still smaller than one.
But this number is still not the largest because zero-point-nine-nine-nine is larger.
But this is still not the maximum since zero-point-nine-nine-nine-nine is larger.
No matter how many nines we keep tacking on
we will never reach a maximum number in the interval.
No matter which number we pick there will always be a number closer to one.
The fact is, that there is no maximum number in an open interval.
This same logic explains why it is also impossible to find a minimum number in an open interval.
However, regardless of whether an interval's endpoints are included or excluded
they still bound the interval to a finite length on the number line.
But what if an interval has no upper or lower bound?
There are three types of unbounded intervals.
If an interval has no upper bound, we say that the interval is "right-unbounded".
In interval notation, this is indicated by replacing the right endpoint
with an infinity symbol.
In set-builder notation, this is the set of all x
such that x is greater than or equal to negative one.
If an interval has no lower bound, we say that the interval is "left-unbounded".
This is indicated by replacing the left endpoint with a negative infinity symbol.
This is the set of all x
such that x is less than or equal to one.
And if an interval has neither an upper nor lower bound
we say that the interval is "unbounded on both ends".
This interval therefore includes the entire number line
and is equal to the set of real numbers.
In interval notation, an infinity symbol is always enclosed with a left or right paren.
This is because the infinity symbol does not represent a number.
It is a symbol which indicates that the interval goes on forever
in the positive or negative direction.
We therefore use the parentheses
to indicate that the infinity symbol is not a member of the set.
All the intervals we have discussed so far are "proper intervals"
because they include more than one point.
However, it is possible for an interval to include only one point
or no points.
In the next lecture, we will explore these "degenerate intervals"
as well as sets which contain multiple intervals.