1 April 2009 Instructor: Tasneem Darwish 1

University of PalestineFaculty of Applied Engineering and Urban Planning

Software Engineering Department

Formal Methods

Relations

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Outlines

Binary relations.

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Introduction

In a formal specification, it is often necessary to describe relationships between Objects.

Example:1.This object is greater than that.2.This action takes priority over that one.

These relationships can be described using simple mathematical objects called relations.

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Binary RelationsWe can define relations between a finite number of objects, but

its enough to define binary relations.

Binary relations are relations that express links between pairs of objects.

In our mathematical language a relation is a set of ordered pairs, a subset of a Cartesian product.

If X and Y are sets, then X ↔Y denotes the set of all relations between X and Y.

The relation symbol can be defined using generic abbreviation:

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Binary RelationsExample 7.1 The set of relations {a, b} ↔ {0, 1} is the set of sets of pairs

The relation that associates a with 0, a with 1, and b with 0 is the set

The expression is another way of writing (x,y)

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Binary RelationsExample 7.2 The relation drives is used to record which makes of car are driven by the members of a small group of people.

If the group of people is defined by:

and the choice of cars is defined by

Then drives is an element of , and the statement `Kate drives a cortina' could be formalised as:

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Binary RelationsWe may also choose to introduce a relation as an infix symbol: a

symbol that sits between its arguments.

Many familiar relations are written in this way:the less-than-or-equal-to relation on numbers; the subset relation on sets.

If the pair (x, y) is an element of the infix relation R, then we may write x R y.

When we define an infix relation, we include underscores to indicate where the arguments should go.

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Binary RelationsExample 7.3 The relation drives could be defined by

•Simple relations can be illustrated using diagrams with arrows, or graphs.

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Binary RelationsExample 7.3 The relation drives could be defined by

•Simple relations can be illustrated using diagrams with arrows, or graphs.