Ways of counting... Review of Basic Counting Principles. Spring Ways of counting... Spring / 11

Transcription

1 Ways of counting... Review of Basic Counting Principles Spring 2013 Ways of counting... Spring / 11

2 1 Introduction 2 Basic counting principles a first view 3 Counting with functions 4 Using functions to count Pigeonhole Principle Ways of counting... Spring / 11

3 Introduction Overview: Learning outcomes By the conclusion of today s class we will: Review common counting principles, such as product rule, disjoint addition rule. Describe how to use functions to count: bijections and the Pigeonhole Principle. Distinguish four basic counting applications: permutations (arrangements) with and without replacement, and combinations (order independent), with and without replacement. Additional counting problems will be addressed, for example: counting outcomes in trees. Brief (cursory) understanding of probabilities and their relationship to sets. Ways of counting... Spring / 11

4 Basic counting principles a first view Counting the ways... Suppose that we have two sets of items, S and T. Assume, further, that A and B are disjoint, meaning A B =. How many ways can I choose elements from S or T? Ways of counting... Spring / 11

5 Basic counting principles a first view Counting the ways... Suppose that we have two sets of items, S and T. Assume, further, that A and B are disjoint, meaning A B =. How many ways can I choose elements from S or T? S T = S + B. Why? Ways of counting... Spring / 11

6 Basic counting principles a first view Counting the ways... Suppose that we have two sets of items, S and T. Assume, further, that A and B are disjoint, meaning A B =. How many ways can I choose elements from S or T? S T = S + B. Why? Suppose that we have two sets of items, S and T : How many ways can I choose elements from S and T? Ways of counting... Spring / 11

7 Basic counting principles a first view Counting the ways... Suppose that we have two sets of items, S and T. Assume, further, that A and B are disjoint, meaning A B =. How many ways can I choose elements from S or T? S T = S + B. Why? Suppose that we have two sets of items, S and T : How many ways can I choose elements from S and T? S T = S T. Why? Ways of counting... Spring / 11

8 Basic counting principles a first view Counting the ways... Suppose that we have two sets of items, S and T. Assume, further, that A and B are disjoint, meaning A B =. How many ways can I choose elements from S or T? S T = S + B. Why? Suppose that we have two sets of items, S and T : How many ways can I choose elements from S and T? S T = S T. Why? The first property is often called the sum rule, and the second is called the product rule. Ways of counting... Spring / 11

9 Counting with functions Sets of functions Usually given as a definition, the exponent set is the total number of functions (maps) possible between two sets: Definition (Exponent set) The exponent set of f : A B is the number of maps from A to B, {f : A B}, which is #(B) #(A) or, alternatively written as B A. Problem Show how the definition of the exponent set follows from the product rule. Ways of counting... Spring / 11

10 Counting with functions Generalizing the sum rule In the original statement of the sum rule, we assumed that sets S and T were disjoint. What if they are not? Problem Assume that we are manufacturing license plates. Ways of counting... Spring / 11

11 Counting with functions Generalizing the sum rule In the original statement of the sum rule, we assumed that sets S and T were disjoint. What if they are not? A B A B = A + B A B. Problem Assume that we are manufacturing license plates. Ways of counting... Spring / 11

12 Counting with functions The Characteristic or Indicator function Another special function associated with every set is Definition (Characteristic function) Let S T, then the characteristic function χ S : T {0, 1} maps every point in S that appears in T to 1, or to 0 otherwise: { 1 x T χ S (x) = for all x T. 0 x T, Problem Count the maps described by χ in terms of #(S). Use the definition of Exponent Sets. Construct a bijection to Power Sets. Construct a bijection to n-place binary integers. Ways of counting... Spring / 11

13 Using functions to count Counting elements by counting functions Complex counting can often be simplified by counting functions. We ve already seen examples: retractions, sections, and bijections tell us something about the cardinalities of their domains and codomains. Exponent sets also tell us something about generalized functions. Characteristic functions relate the number of subsets to the number of elements in the domain and codomain of a function. Let s simplify and generalize counting strategies from these. Ways of counting... Spring / 11

14 Using functions to count Simplifying assumptions... Consider the function e : A A, which is a special map, called an endomap, hence its name e. Assume that A is non-empty and is finite. Using the product rule, how many bijections can we find from A to A? Ways of counting... Spring / 11

15 Using functions to count Bijections, revisited. Recall that bijections preserve the properties of their domains and codomains. In the case of abstract sets, if f : A B has an inverse, say f 1 : B A, then A = B, or A is isomorphic to B. Then #(A) = #(B), or A = B. f and its inverse are unique. f and its inverse are both injective (one-to-one) and surjective (onto). But, we can define N-to-1 relationships and then bijections can count multiples. Problem Construct a 2-to-1 bijection from students in this class to N. Ways of counting... Spring / 11

16 Using functions to count Pigeonhole Principle The Pigeonhole Principle Sometimes common sense takes us a long way... Definition (Pigeonhole principle) Given any function f : A B, if #(A) > #(B), then at least two elements of A must map to one point in B. Often, this principle is stated computationally: given k pigeons and n pigeonholes, the at least one pigeonhole must contain at least n/k pigeons. Example Suppose that I have a drawer containing a dozen socks, some blue, some black. How many socks must I select to ensure that I have a least one pair of matching socks? Ways of counting... Spring / 11