13.2 Intervals
The interval from \( x_1 \) to \( x_2 \), written as
\[
\begin{align}
i &: \Interval_{x}
\\
i &= [x_1, x_2]_{x}
\end{align}
\]
consists of all \( x \)-values from a number \( x_1 : \Number \) to a number \( x_2 : \Number \).
We draw an interval with an orientation to indicate the interval's direction. The interval \( [3, 5]_{x} \) is pictured on the left, and the interval \( [5, 3]_{x} \) is pictured on the right.
Negating an interval reverses the orientation.
\[
\begin{align}
- [x_1, x_2]_{x} &= [x_2, x_1]_{x}
\end{align}
\]
So
\[
\begin{align}
- [3, 5]_{x} &= [5, 3]_{x}
\end{align}
\]
for example.
Boundary
Let's define the boundary operator, \( \boundary \), for blips and intervals. The boundary of any blip is zero.
\[
\begin{align}
\boundary [x_0]_{x} &= 0
\end{align}
\]
And the boundary of an interval is a difference of blips.
\[
\begin{align}
\boundary [x_1, x_2]_{x} &= [x_2]_{x} - [x_1]_{x}
\end{align}
\]
We use positive and negative signs on blips as orientations:
-
a positive blip, \( + [x_2]_{x} \), represents an arrival, and
-
a negative blip, \( - [x_1]_{x} \), represents a departure.
The orientation of an interval determines the orientations on the boundary.
Example.
Let's picture the interval
\[
\begin{align}
[1, 4]_{x} &: \Interval_{x}
\end{align}
\]
and its boundary
\[
\begin{align}
\boundary [1, 4]_{x} &= [4]_{x} - [1]_{x}.
\end{align}
\]
We've labeled each blip with a positive or negative sign to record its orientation.
