Pigeonhole principle, math, Marketing Management

A binary relation R from A to B is said to be a function if for every element a in A, there is a unique element b in B so that (a, b) is in R. For a function R from A to B, instead of writing (a, b) ? R, we also use the notation R(a) = b, where b is called the image of a. the set A is called the domain of the function R, and the set B is called the range of the function R. The notion of a function is but a formalization of the notion associating or assigning an element in the range to each of the elements in the domain. For example, let A be a set of houses and B be a set of colours. Then a function from A to B is an assignment of colours for painting the house. Following the convention of representing a binary relation in tabular form that was introduced earlier, we can represent the function, where the left column contains all the elements in the domain and the right column contains their corresponding images.

A function from A to B is said to be an onto function if every element of B is the image of one or more elements of A. A function from A to B is said to be a one-to-one function if no two elements of A have the same image. A function from A to B is said to be one-to-one onto function if it is both onto and a one-to-one function. Let A be a set of workers and B1B2 and B3 be sets of jobs. An onto function from A to B1, is an assignment of the workers to the jobs so that every job has at least one worker assigned to it; a one-to-one function from A to B2 is an assignment such that no two workers will have the same job; and a one-to-one onto function from A to B3 is an assignment such that every job has a worker assigned to it, and no two workers are assigned to the same job.

In the literature, an onto function is also called a surjection, a one-to-one function is also called an injection, and a one-to-one function is also called a bijection.

A well-known proof technique in mathematics is the so-called pigeonhole principle, also known as the shoe box argument or Dirichlet drawer principle. In an informal way the pigeonhole principle says that if there are “many” pigeons and “a few” pigeonholes, then there must be some pigeonhole occupied by two or more pigeons, and then there must be some pigeonhole occupied by two or more pigeons. Formally, let D and R be finite sets. If |D| > |R|, then for any function ƒ from D to R, there exist d1, d2 ? D such that ƒ(d1) = ƒ(d2). Some trivial applications of the pigeonhole principle are: Among 13 people, there are at 2 of them who were born in the same month. Here the 13 people are the pigeons and the 12 months are the pigeonholes. Also, if 11 shoes are selected from 10 pairs of shoes there must be a pair of matched shoes among the selection. Here the 11 shoes are the pigeons and 10 pairs are the pigeonholes. The pigeonhole principle can be stated in slightly more general form: For any function ƒ from D to R, there exist i elementsd1, d2, … di in D, i = |D|/|R|, such that ƒ(d1) = ƒ(d2) = ƒ(di)

Posted Date: 2/14/2012 1:44:08 PM | Location : United States

Related Discussions:- Pigeonhole principle, math, Assignment Help, Ask Question on Pigeonhole principle, math, Get Answer, Expert's Help, Pigeonhole principle, math Discussions

Write discussion on Pigeonhole principle, math
Your posts are moderated
Related Questions
Marketing implementation: Planning goods strategies is only a start toward successful marketing. A brilliant marketing strategy counts for little if the company fails to imp

Question 1: "Various factors have created a need for businesses to move from a Traditional Marketing approach to a Relationship Marketing one." (a) Describe this statement.

How should Shoppers’ Stop develop its demand forecasts?

Ask question #Minimum 100 words accept 1. Based on the segmentation variables discussed in the chapter, construct a profile for Vanilla Bicycle’s probable target market. ed#

State about the Demand - pricing decisions Demand also affects the price in a big way.  When there is limited supply of a product and demand is high, people buy even if hig

Two events A and B are said to be dependent when  B  can occur only when A is known to have occurred (or vice versa). The probability attached to such an event is called the condit

what is the square root of 9?

A Convertible Ultrabook is a new type of notebook which functions both as a laptop and tablet mode when required. Ultrabooks are designed to feature reduced bulk without compromisi

Enumerate about the Non-durable Goods Non-durable goods are products which are normally consumed in one go or last for a few uses. Illustrations of such products are soap, s