Specification of Reactive Hardware/Software Systems - Electronic ...

7.4 Mathematical Preliminaries 235
£
0¥ 1¥ 2¥¡ ¡ ¡ ¦
£
¡¢ ¡ 0¦
are called the elements of the set. These elements may be numbers, functions, strings or
they can be sets themselves. For instance the set
denotes an infinite
collection of natural numbers. The curly braces are used to ’enumerate’ the elements
of the set. The three dots indicate that the partial list of numbers should be completed
to define the infinite collection of natural numbers. Another way to specify a set is by
referring to other sets and to properties that elements may or may not have. For instance
n n mod 2 defines the set of even natural numbers. An alternative way to
define this set is by £
¦ ¡£ ¡£ ¡
¡£ ¡
¤
2n n . Here n denotes that n is an element of . If this is
not true we write n . The empty set, i.e. the set that contains no elements, is denoted
by . A set, such as £
0¦ , that contains only one element is called a singleton set.
¥
¡ ¥
¦ §
¨
©
Operators and Relators
Given two sets V and W we will write V W to indicate that V is a subset of W. This
means that each element of V is also element of W. V W indicates that V is not a
subset of W. The union of V and W is denoted by V W and their intersection by V W.
Sometimes we will take a generalised union of sets. If V is a set of sets, then V denotes
the union of all sets that are element of V. Finally,
is the collection of all subsets of V.
(V) denotes the powerset of V. This
Typical Elements
In Chapters 8 and 9 we will define quite a number of different collections. To refer
to elements of these collections, the collections themselves also have to be specified.
However, this would not improve the readability. Therefore we follow the convention
to use typical elements of a collection. If V is a collection we could introduce v¥ w¥¡ ¡ ¡ as its
typical elements. This means that v and w (but also v1¥ v2 ¥ w1, etcetera) refer to arbitrary
elements of V. Instead of saying that v¥ w¥¡ ¡ ¡ are typical elements of V, we also say that
v¥ w¥¡ ¡ ¡ range over V.
Syntactic Elements and Meta Variables
To define the syntax of a language one has to introduce collections of syntactic elements.
Syntactic elements are the syntactic representations of statements, expressions, variables,
specifications, etcetera. The typical elements of a collection of syntactic elements are
called its syntactic meta-variables. The term meta indicates that these variables themselves
are not part of the collection. They are only used to refer to these elements. In principle
the same holds for the typical elements of an arbitrary collection. However, within the
context of the definition of a language the distinction is important to make. For arbitrary
sets the distinction is and can often be blurred.
7.4.2 Cartesian Products and Binary Relations
Cartesian Products and Ordered Tuples
The cartesian product of two sets V and W, denoted by V W, is the set of all ordered pairs
v¥ w with v ¡ V and w ¡ W. The pair v¥ w is called an ordered tuple. More generally,
if v1¥¡ ¡ ¡ ¥ vn are n objects, not necessarily distinct, then v1¥¡ ¡ ¡ ¥ vn is called an ordered
n-tuple, or n-tuple for short. Ordered pairs are the same as ordered 2-tuples. If V1¥¡ ¡ ¡ ¥ Vn
are n collections, then the n-fold cartesian product V1 ¡ ¡ Vn is the set of all ordered