A beautiful application is provided by
.
Example Let 
denote a
finite set and let 
be any given properties. We want to express the number of elements of
which have none of these properties in terms of numbers of elements which
have some of these properties. In order to do this we indicate by 
the fact that 
has the property 
, and for an index set 
we put
Furthermore we put
and it is our aim to express 
in terms of the 
.
The set on which we shall define an involution is
A disjoint decomposition of this set is 
, where
Now we introduce, for 
, the number 
, and define 
on 
as follows:
Obviously 
provided that 
. Furthermore 
and is sign-reversing, so that from 
we obtain
Thus we have proved
.
The Principle of Inclusion and Exclusion
Let
be a finite set and any properties, while 
denotes
the set of elements of having each of the properties 
. Then the order of the subset 
of elements having none of these properties is
A nice application is the following derivation of a formula for the
values of the Euler function 
We would like to express
in terms of the different prime divisors
of 
In order to do this we put 
and we define 
to be the property divisible by 
,
obtaining from the Principle of Inclusion and Exclusion the following
expression for
the set of elements of 
which are relatively prime to 
which gives the desired expression for the value of the Euler
function at
Exercises
.