Maps form a subclass of sets.
- A map is a set that is either empty or whose elements are all
ordered pairs.
An ordered pair is a tuple whose first two components and no others are
defined.
- There are two special operators for evaluating a map at a point in its
domain.
Suppose that
F
is a map.
F(EXPR)
will evaluate to the value of the second component
of the ordered pair whose first component
is the value of
EXPR
, provided there is exactly
one such ordered pair in
F
;
otherwise, it evaluates to
OM
.
F{EXPR}
will evaluate to the set of all values
of second components of ordered pairs in
F
whose first component is the value of
EXPR
.
If there are none such, its value is the empty set.
- A map in which no value appears more
than once as the first component of an
ordered pair is called a single-valued map or smap;
otherwise, the map is called a multi-valued map or mmap.