In this introductory chapter some mathematical notions are presented rapidly

empty set, the set containing no elements. It is usually denoted by the symbol
, so All other subsets of are proper and non-empty.
Consider for instance } as ambient set. Then

Note that contains 3 elements (it has cardinality 3), while ) has
elements, hence has cardinality 8. In general if a finite set (a set with a finite
number of elements) has cardinality n, the power set of X has cardinality .
Starting from one or more subsets of , one can define new subsets by means
of set-theoretical operations. The simplest operation consists in taking the complement:
if is a subset of , one defines the complement of to be the
subset

made of all elements of not belonging to (Fig. 1.1, right).
Sometimes, in order to underline that complements are taken with respect to
the ambient space , one uses the more precise notation . The following
properties are immediate:

For example, if and A is the subset of even numbers (multiples of 2), then
is the subset of odd numbers.
Given two subsets and 5 of , one defines intersection of and the
subset
}
containing the elements of that belong to both and , and union of and
B the subset

made of the elements that are either in or in (this is meant non-exclusively,
so it includes elements of ), see Fig. 1.2.
We recall some properties of these operations.
i) Boolean properties:


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ii) commutative, associative and distributive properties:




,

iii) De Morgan laws:

Notice that the condition is equivalent to
There are another couple of useful operations. The first is the difference between
a subset A and a subset B^ sometimes called relative complement of