In this introductory chapter some mathematical notions are presented rapidly

in

(read ' minus '), which selects the elements of that do not belong to . The
second operation is the symmetric difference of the subsets and

which picks out the elements belonging either to or , but not both (Fig. 1.3).
For example, let be the set of even numbers and
the set of natural numbers smaller or equal than 10. Then
is the set of odd numbers smaller than 10, is the set of even numbers larger
than 10, and is the union of the latter two.4bet

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1.2 Elements of mathematical logic
In Mathematical Logic a formula is a declarative sentence, or statement, the truth
or falsehood of which can be established. Thus within a certain context a formula
carries a truth value: True or False. The truth value can be variously represented,
for instance using the binary value of a memory bit (1 or 0), or by the state of
an electric circuit (open or close). Examples of formulas are: '7 is an odd number'
(True), '3 > 12' (False), 'Venus is a star' (False), 'This text is written in english'
(True), et cetera. The statement 'Milan is far from Rome' is not a formula, at least
without further specifications on the notion of distance; in this respect 'Milan is
farther from Rome than Turin' is a formula. We shall indicate formulas by lower
case letters . . . .
1.2.1 Connectives
New formulas can be built from old ones using logic operations expressed by certain
formal symbols, called connectives.
The simplest operation is called negation: by the symbol (spoken 'not p')
one indicates the formula whose truth value is True if p is False, and False if p
is True. For example if is a rational number', then is an irrational
number'.
The conjunction of two formulas p and q is the formula ('p and Q'),
which is true if both p and q are true, false otherwise. The disjunction of p and
q is the formula the disjunction is false if and are both false,
true in all other cases. Let for example is a rational number' and
is an even number'; the formula p⋀ q=’7 is an even rational number' is false since
q is false, and is rational or even' is true because is true.
Many statements in Mathematics are of the kind 'If is true, then is true',
also read as 'sufficient condition for to be true is that p be true', or 'necessary
condition for p to be true is that be true'. Such statements are different ways
of expressing the same formula (' implies or 'if then ), called