The principle of compensation and its application

THE PRINCIPLE
OF
COMPENSATION
AND ITS
APPLICATION


Group members
NG. Duyên

Thu Trang

Hoa Mai


I. COMPENSATION
1. Formula 1: PRINCIPLE
Given 2 sets A, B. We have:


2. FORMULA 2:
Given X set and n is a subset

In which:

We have:


3.THE PRINCIPLE OF COMPENSATION (SIEVE'S
FORMULA) IS:

II. PRINCIPLE OF
GENERALIZED
COMPENSATION
DEFINITION
THEOREM


1.DEFINITION
Consider m objects
. These objects are
respectively attached to weights which are the
elements of some commutative ring K. Each
given object may or may not have properties .
where
Symbol is
=, the sum of the weights of all objects
with properties
M(r) is the sum of the weights of all objects with
exactly r properties.
is the sum of the weights of all objects with no
less than r properties.


2. THEOREM

M(r)= ,

for all r = 0.1,...,n

III-GENERAL
COMPENSATION
FORMULA


Suppose is a subset containing elements of
property , the number of elements with all
properties , , …is denoted by N(, …). Writing the
above quantities over the sets we have: = N( , …)
If the number of elements that have no
properties among n properties , , …, is denoted by
N(…) and the number of elements in the given set
is N, then we deduce that: N( …) = N On the other hand, by the principle of
compensation we have :
= - + –…+

Hence,


IV. APPLICATIONS THE PRINCICPLE
OF
COMPENSATION
TO
SOLVE
THE PROBLEM OF EXTENDING THE COMMON VENN
DIAGRAM BY THE PRINCIPLE
OF
OFFSET
EXERCISES.

NUMBER COUNTING PROBLEM SATISFYING ARITHMETIC
PROPERTIES
THE PROBLEM OF COUNTING THE NUMBER OF INTEGER
SOLUTIONS
BERNOULLI – EULER
PROBLEM
SURJECTIVE COUNT MATH PROBLEM

6

THE PRINCIPLE OF COMPENSATION COMBINES
WITH THE MAPPING METHOD


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Listening.