II. Notes on Data Structuring * - Cornell University

124 c.A.R. HOARE
those values as members. As in the case of the Cartesian Product, the type
name is used as the transfer function, but for sets, the number of arguments
is variable from zero upwards. For example"
primary colour (red, yellow) i.e. orange
liftcall (ground) i.e. only a single button has been
pressed
statusword ( ) i.e. no exception condition.
The last two examples illustrate the concept of a unit set (which must be
clearly distinguished from its only member) and the null or empty set, which
contains no member at all. If the type name is omitted in this construction,
curly brackets should be used instead of round ones in the normal way.
The converse of the null set is the universal set, which contains all values
from the base type. This may be denoted
T. all.
However, this universal set exists as a storable data value only when the base
type is finite.
The basic operations on sets are very familiar to mathematicians and
logicians.
(1) Test of membership: If x is in the set s, the Boolean expression "x in s"
yields the value true, otherwise the value false.
(2) Equality: two sets are equal if and only if they have the same members.
(3) Intersection" sl ^ s2 contains just those values which are in both sl
and s2.
(4) Unions" sl v s2 contain just those values which are either in sl or s2,
or both.
(5) Relative complement" s 1 - s2 contains just those members of sl which
are not in s2.
(6) Test of inclusion: s l c s2 yields the value true whenever all members
of s l are also members of s2, and false otherwise.
(7) The size of a set tells how many members it has.
If the domain type of a set has certain operators defined upon it, it is often
useful to construct corresponding operations on sets. In particular, if the
domain type of a set is ordered, the following operators apply:
(8) rain (s) the smallest member of s; undefined if s is empty.
(9) x down n is a set containing just those values whose nth successors are
in s.
(10) x up n is a set containing just those values whose nth predecessors
are in s.