II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 125
(11) Range (a, b) is the set containing a, succ(a),..., b if a ~< b, and which
is empty otherwise.
The most useful selective updating operations on sets are:
x'v y; join the set y to x
x:v T(a) add the member a to x
x: ^ y; exclude from x all members which are not also members
ofy
x:- y exclude from x all members which are also members
ofy
x: down n subtract n from every member ofx and exclude members
for which this is not possible
x:up n add n to every member of x, and exclude members for
which this is not possible
It is also sometimes useful to select some member from x and simultaneously
remove it from x. This operation can be expressed by the notation"
a from x.
If the domain type of x is ordered, it is natural that the selected member
should be the minimum member of x; otherwise the selection should be
regarded as arbitrary.
It is often useful to define the value of a set by giving some condition B
which is satisfied by just those values of the domain type which are intended
to be members of the set. This may be denoted:
{i:OrB}
where i is a variable of type D regarded as local to B,
and B is a Boolean expression usually containing and depending on i.
In order for this expression to denote a value of the powerset type it is
essential that the cardinality of D be finite, and that B is defined over all
values of the type.
Finally, it is frequently required to perform some operation on each
member of some set, that is to execute a loop with a counting variable which
takes on successively all values in the set. A suitable notation for expressing
this is:
for x in s do...
If the base type of s is an ordered type, it seems reasonable to postulate that
the elements will be taken in the natural order, starting with the lowest.
For an unordered base type, the programmer does not care in which order
the members are taken, and he leaves open the option to choose an order
that contributes best to efficiency.