# 03 - Fall 2009 CS131 Combinatorial Structures Homework 3...

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Fall 2009 CS131 – Combinatorial Structures Homework 3 Homework 3, due Sept 29 You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check ex- ercises of the easier kind in the book, try to solve them, too! Problem 1. (10pt) Let A , B be ﬁnite sets, | A | = m > 0, | B | = n > 0. Express the number of injective functions from A to B in terms of m and n . When is this number 0? Solution. If m > n then this number is 0, one cannot map a smaller set into a larger one injectively. Otherwise, it is n ( n - 1) ··· ( n - m + 1). Indeed, let f ( x ) be the function. If A = { a 1 ,..., a m }, then there are n choices for f ( a 1 ), then to keep injectivity, n - 1 choices for f ( a 2 ), then n - 2 choices for f ( a 3 ), and so on. Problem 2. (10pt) Let A , B be ﬁnite sets, | A | = m > 0, | B | = 2. Express the number of surjective functions from A to B in terms of m . Solution. There are 2 m functions from A to B = { b 1 , b 2 }, but some of them take only one value, not two, so they are surjective. In fact there are just two functions taking

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## This note was uploaded on 10/15/2011 for the course MATHS 100 taught by Professor Fredphelps during the Fall '11 term at Jordan University of Science & Tech.

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03 - Fall 2009 CS131 Combinatorial Structures Homework 3...

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