Applied Combinatorics With Problem Solving by Bradley Jackson, Dmitri Thoro

By Bradley Jackson, Dmitri Thoro

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Show that R is an equivalence relation and deduce that there is a one-to-one correspondence between partitions of X and equivalence relations on X. 19. Suppose 5 points are chosen at random inside an equilateral triangle with sides of length 1. Show that there is at least one pair of these points that are separated by a distance of at most 1/2. Hint: Divide the triangle into 4 suitable regions. 20. Suppose that f is a function from A to B, where A and B are finite sets with Al = IBI. Show that f is an injection if and only if f is a surjection.

The following principle is often used to solve a slightly different kind of problem, which might be classified as an existence problem. That is, under certain circumstances we are actually able to show that a particular kind of arrangement must always exist. This principle can be described most colorfully in terms of an assignment of pigeons to holes. THE PIGEONHOLE PRINCIPLE If n + 1 or more pigeons are assigned to n holes, at least two pigeons are assigned to the same hole. 4 35 FUNCTIONS The pigeons can be thought of as the objects of one set A and the holes as another set B of possible images.

Prove that the check digit will always be changed. Hint: If ab ac (mod n), where a and n are relatively prime, then b c (mod n). c) Show that when two digits are miscopied, the error may not be detected. d) Prove that if two adjacent digits are interchanged, the check digit will change, indicating an error. e) What will happen to the check digit if two nonadjacent digits are interchanged? 2. 4 FUNCTIONS 1ff is a function from a set A to a set B, we write f: A —+ B. In terms of ordered pairs, the function f can be thought of as a set of ordered pairs in A x B, where each a in A is the first coordinate of exactly one ordered pair.

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