I give an introduction to algorithmic uses of the principle of inclusion–exclusion. The presentation is intended to be be concrete and accessible, at the expense of generality and comprehensiveness. R T 1 The principle of inclusion–exclusion. There are as many odd-sized as even-sized subsets sandwiched between two different sets: For R ⊆ T ,

I give an introduction to algorithmic uses of the principle of inclusion–exclusion. The presentation is intended to be be concrete and accessible, at the expense of generality and comprehensiveness. R T 1 The principle of inclusion–exclusion. There are as many odd-sized as even-sized subsets sandwiched between two different sets: For R ⊆ T ,

Although the following proof is implicit in Garsia and Milne's paper [I], it is scj elegant that we felt that it should be presented by itself for the benefit of the general mathematical public. The idea behind the proof was further exploited by Remmel [2] and Wilf [3]. Consider a set A of elements each of which possess a (possibly emptyj subset of the properties (1,. .. , n}. The inclusion-exc...

P. Grzegorzewski [3] has worked the probability version of the inclusion-exclusion principle and made a generalization for IF-events. He had applied two versions of the generalized formula, corresponding to different t-conorms and so defined the union of IF-events. This paper contains the generalization of the Grzegorzewski theorem. We prove it for mappings from the set of IF sets to the unit i...

The aim of this work is to describe, at the quantum level, a many body system obeying to a generalized Exclusion-Inclusion Principle (EIP) originated by collective effects, the dynamics, in mean field approximation, being ruled by a nonlinear Schrödinger equation. The EIP takes its origin from a nonlinear kinetic equation where the nonlinearities describe interactions of different physical natu...

Using an operator approach we derive Sylvester-Whitworth formulae for sets A'a. By the same token we treat the problem where both sets of A's and B's are involved. Our result extends the Sylvester-Whitworth inclusion and exclusion formula to the resolution of the number of elements in exactly mi sets of A's and mi sets of B's respectively. The formula are applied to the complete graph and compl...

where U = {1, 2, . . . , n}. Example. Suppose we ask how many integers less than 120 are relatively prime to it. Since 120 has prime factors 2, 3, and 5, any number that is relatively prime to it is not a multiple of any of these. We can solve this using the Inclusion-Exclusion Principle as follows. Let A = {1, 2, . . . , 120}, the set of integers from 1 to 120. Let S1 be the multiples of 2 in ...