In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used.
The rule of sum, rule of product, and inclusion–exclusion principle are often used for enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same number of elements. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context.
Many combinatorial identities arise from double counting methods or the method of distinguished element. Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations.
Rule of sum
The rule of sum is an intuitive principle stating that if there are a possible outcomes for an event (or ways to do something) and b possible outcomes for another event (or ways to do another thing), and the two events cannot both occur (or the two things can't both be done), then there are a + b total possible outcomes for the events (or total possible ways to do one of the things). More formally, the sum of the sizes of two disjoint sets is equal to the size of their union.
Rule of product
The rule of product is another intuitive principle stating that if there are a ways to do something and b ways to do another thing, then there are a · b ways to do both things.
Inclusion–exclusion principle
The inclusion–exclusion principle relates the size of the union of multiple sets, the size of each set, and the size of each possible intersection of the sets. The smallest example is when there are two sets: the number of elements in the union of A and B is equal to the sum of the number of elements in A and B, minus the number of elements in their intersection.
Generally, according to this principle, if A1, ..., An are finite sets, then
Rule of division (combinatorics)
The rule of division states that there are n/d ways to do a task if it can be done using a procedure that can be carried out in n ways, and for every way w, exactly d of the n ways correspond to way w.
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Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n.
In mathematics, the term combinatorial proof is often used to mean either of two types of mathematical proof: A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. Since those expressions count the same objects, they must be equal to each other and thus the identity is established. A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.
Couvre les identités algébriques, la trigonométrie et les fonctions réelles, y compris les fonctions injectables, surjectives, bijectives et réciproques.
Vision systems built around conventional image sensors have to read, encode and transmit large quantities of pixel information, a majority of which is redundant. As a result, new computational imaging sensor architectures were developed to preprocess the r ...
EPFL2023
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if logr/logs is irrational and X and Y are ×r- and ×s-invariant subsets of [0,1], respectively, then $\dim_\text{ ...
2021
In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where d = 0. Their values have already been computed by Kim and Stanton in 2015, h ...