The document discusses the mathematical foundation of computer science, specifically the inclusion-exclusion principle. It provides an overview of the principle, which is a counting technique for obtaining the number of elements in the union of finite sets. It describes how the principle works for two sets using a formula, and how it generalizes to three or more sets. The principle involves including the cardinalities of sets, excluding the intersections of pairs of sets, and continuing this pattern of inclusion and exclusion for higher-order intersections as needed. Some applications of the principle in discrete mathematics and combinatorics are then outlined.

1. MATHEMATICAL FOUNDATION OF
COMPUTER SCIENCE
by,
S.Subhalakshmi
M.Sc(Computer Science)
Nadar saraswathi college of arts and science.

2. Inclusion and Exclusion Principle:
The Inclusion and Exclusion Principle is a counting technique
which generalizes the familiar method of obtaining the number of elements in
the union of two finite sets, symbolically expressed as
where A and B are two finite sets and |S| indicates the cardinality of a set
S

3. The principle is more clearly seen in the case of three sets, which for the set A , B
and C is given by,
Generalizing the results of these examples gives the principle of inclusion–exclusion
.To find the cardinality of the union of n sets:

4. 1. Include the cardinalities of the sets.
2. Exclude the cardinalities of the pairwise intersections.
3. Include the cardinalities of the triple-wise intersections.
4. Exclude the cardinalities of the quadruple-wise intersections.
5. Include the cardinalities of the quintuple-wise intersections.
6. Continue, until the cardinality of then tuple-wise intersection is
included (if n is odd) or excluded (n even).

5.  The name comes from the idea that the principle is based on
over-generous inclusion, followed by compensating exclusion.
 This concept is attributed to Abraham de Moivre (1718), but it first
appears in a paper of Daniel da Silva (1854), and later in a paper
by J. J. Sylvester (1883).
 Sometimes the principle is referred to as the formula of Da Silva,
or Sylvester due to these publications.
“One of the most useful principles of enumeration in discrete
probability and combinatorial theory is the celebrated principle of
inclusion–exclusion. When skillfully applied, this principle has yielded
the solution to many a combinatorial problem."

6. In its general form, the principle of inclusion–exclusion states that for
finite set A1, ..., An one has the identity
This can be compactly written as

7. or
In Demorgan’s law we have,
let P1, P2, ..., Pn be a list of properties that elements of a set S including
may or may not have the principle of inclusion and exclusion.

8. EXAMPLES:
Counting integers:
How many integers in {1,...,100} are not divisible by 2, 3 or 5?
Let S = {1,...,100} and p1 is divisible by 2,p2 is divisible by
3,p3 is divisible by 5.
|A1| = 50, |A2| = 33, and |A3| = 20.
16 is divisible by 6,10 is divisible by 10,6 is divisible by 15.
There are three integers divisible by 30.The number of integers not
divisible by 2,3,5.
100 − (50 + 33 + 20) + (16 + 10 + 6) − 3 = 26.

9. Applications:
1.counting derangements:
It is a combinatorial problem of counting all derangements
of a finite set. The derangements has no fixed points.
2. Counting intersections:
De Morgan's law, can be used to count the cardinality of the
intersection of sets as well.
3. Graph coloring:
A well known application of the principle is the construction
of the chromatic polynomial of a graph.

10. 4. Bipartite graph perfect matchings:
The number of perfect matchings of a bipartite graph can be calculated using
the principle.
5.Number of onto functions:
The Principle of inclusion–exclusion gives the number of onto functions
between A and B as:
6.Permutations with forbidden positions:
A permutation of the set S = {1,2,...,n} where each element of S is restricted to
not being in certain positions is called a permutation with forbidden positions.