Structure of Z notation of formal methods

1. Formal methods in
software engineering
Lecture on Structure:Element of Z
Prof Engr. Faiz ul haque Zeya

2. Topics
 Structure:
 Tuple,
 Records,
 Relations,
 Tables,
 Databases,
 Pairs
 Binary Relations,
 Functions,
 Sequences and
 Operators

3. Tuple
 Tuples associate particular elements of any type, in a fixed order
 If we use set (2,11,11) and (11,2,11) won’t be different.
 Tuples are instances of Cartesian product types, sometimes called cross
product types. We define product types using the cross product symbol x. This
abbreviation definition introduces a set of tuples named DATE
 DATE=DAY X MONTH X YEAR
 DATE has product type P(Z X Z X Z)

5. Relation, Table and Database
 A set of tuples is called a relation. Relations can model tables and databases.
You have probably heard of the relational database, which is just a database
where the data are stored in one or more relations

7. Pair and binary relation
 It has two components.
 . We can use a pair to associate a name with a telephone extension number,
as in (aki, 4117). Z provides an alternate syntax for pairs that uses the maplet
arrow ---> to emphasize the asymmetry between the two components. The
pair (aki, 4117) can also be written aki -->4117
 Z also provides the first and second operators for extracting each component
from a pair: first(aki,41) = aki second(aki,41) = 41 operators like these that
extract components from structures are called projection operators.

9. Domain and range

10. Operators for relations
The relation image operator
 The relational image operator can model table lookup. Its first argument is a
relation, its second argument is a set of elements from the domain, and its
value is the set of corresponding elements from the range. It is notated in an
unusual mixfix syntax: Thick brackets (]...[) surround the second argument.
To look up the numbers for Doug and Philip in the phone relation, we use the
relational image: phone

11. Domain restriction operator

12. Range restriction operator

13. Combination of domain and range
restriction

14. Anti restriction operator

15. Override operator

17. Inverse operator

18. Composition relation
 Relational composition formalizes this kind of reasoning: It merges two
relations into one by combining pairs that share a matching component.

19. Binary relation and linked data
structure.
 Relations are not just for modelling tables and flat databases. They can model
linked data structures as well. Linked data structures are often pictured as
graphs: networks of nodes connected by arcs. Data flow diagrams, state
transition diagrams, and syntax trees are all examples of graphs.

21. Function
 Sometimes we need to associate a single item with each element in a set. For
this we use a special kind of relation, called & function. A function is a binary
relation where an element can appear only once as the first element in a pair.

23. Function application

24. Partial and total function
 The domain of a partial function might not include every element of the
source set.
 Total function apply to every element of the source set
 Square of x.

25. Injections
 every element of the function's codomain is the image of at most one element
of its domain.

26. SEQUENCES
 Sequences are ordered collection

27.  Another way to declare sequence is:

28. Operators