The span of two intervals and
, denoted
, is the smallest interval whose extension contains the
extensions of both
and
.
For example, the span of (1,4) and [2,6] is (1,6].
Similarly, the span of [3,7) and (3,7] is [3,7].
More generally, the lower endpoint of
is the minimum of the
lower endpoints of
and
.
The lower endpoint of
is open or closed depending on
whether the smaller of the lower endpoints of
and
is open or
closed.
Analogously, the upper endpoint of
is the maximum of the
upper endpoints of
and
.
The upper endpoint of
is open or closed depending on
whether the larger of the upper endpoints of
and
is open or
closed.
More precisely,
can be computed as follows:
The notion of span will be used in Section 4.7.
Let us extend the notion of span to two sets of intervals by the following definition:
We will want to compute the span of two sets of intervals and
, when
both
and
are represented as spanning intervals.
Additionally, we will want the resulting span to be represented as a small set
of spanning intervals.
Given two normalized spanning intervals and
, their span
is a set of normalized spanning intervals whose extension
is the span of the extensions of
and
.
One can compute
as follows:
An important property of normalized spanning intervals is that for any two
normalized spanning intervals and
,
contains
at most four normalized spanning intervals.
In practice, however, fewer normalized spanning intervals are needed, often
only one.
The intuition behind the above definition is as follows.
Consider, first, the lower endpoint.
Suppose that the lower endpoints and
of
and
are in
and
respectively.
That means that
and
.
The lower endpoint of
will be
, when
,
and
, when
.
Thus it will be
, for all
, and will be
,
for all
, where
, when
, and
,
when
.
Thus there will be two potential ranges for the lower endpoint of
:
and
.
When the lower endpoint of
is taken from the former, it
will be open or closed depending on whether the lower endpoint of
is
open or closed.
When it is taken from the later, it will be open or closed depending on
whether the lower endpoint of
is open or closed.
Thus the lower endpoint of
can be either
or
.
Analogous reasoning can be applied to the upper endpoints.
If the upper endpoints of
and
are
and
respectively, then there are
two possibilities for the upper endpoint of
, namely
and
, where
, when
, and
, when
.