Statistics for Financial Engineering Session 2: Basic Set Theory March 19 th, 2006 Topics What is a set? Notations for sets Empty set Inclusion/containment and subsets Sample spaces and events Operations on sets: unions, intersections, complements o Associative, commutative, distribution laws o DeMorgan s laws o Infinite unions and intersections The contents of Session 2 may be familiar to you, especially if you have studied higher mathematics. However, it is very important to understand basic set theory and settheoretic operations, as they are the foundation and language of probability theory.
Set theory is the foundation and language of all higher mathematics. Indeed, all mathematical objects integers, real numbers, functions, etc. can be defined and constructed as certain sets, using the basic set-theoretic operations. Of course, we are concerned here with set theory as it is needed for probability theory. We will see that set theory is the language used to define the essential concepts of probability: A sample space is defined as a set. An event is any subset of the sample space. The definition of a probability measure on a sample space includes a notion of additivity, which involves disjoint subsets of the sample space, and a certain property of their union. The notion of conditional probability is defined in terms of an intersection of subsets of the sample space. Thus, to understand the definitions of probability theory, you must be comfortable with the notions of set theory mentioned above. The goal of this lecture are to define these notions of set theory, and provide some illustrative examples. What is a set? A set is simply any collection of objects. The objects are called the members or the elements of the set. The key point about sets is that a set is completely defined by its elements. Often the elements of a given set are mathematical in nature, but that does not have to be the case. A set is any collection of objects. Example: The following are all sets: i. 1,2, 3, 4,5 ii. { 1,2,3,...,999} iii. the set of all positive integers Ν = 1,2,3,... iv. the set of all odd positive integers 1,3,5,7... v. the real numbersr vi. the unit interval 0,1 [ ]= x : x R and 0 x 1 vii. S 1 = H,T viii. S 2 = { HH,HT,TH,TT} ix. S n = the set of sequences of length n of Hs and Ts x. S = the set of infinite sequences of Hs and Ts
Notations for sets We have introduced some basic notation for sets in the examples above. The notation for describing a set is by specifying its members within braces, as above. As we mentioned above, a set is completely defined by its members. Thus, it doesn t matter in which order we list the elements. For example: { 1,2, 3, 4,5}= { 4,1,5,3,2 }= 5,4,3,2,1 { H,T}= { T,H} Let s go through the examples above. In (i), we listed all elements of the set. This is possible because the set is finite. For (ii), it would be possible to list all the members of the set, since it too is finite, but clearly it would be tedious and time-consuming to do so. Instead, we use dots to represent a (finite) number of omitted elements. Examples (ii)-(vi), however, are infinite sets, and so it is impossible to list all of their members. In (iii) and (iv), we again use dots, but here they represent an infinite number of elements. Example (v) consists of the set of real numbers. For our purposes, we will assume the real numbers are given as a mathematical set. 1 In (vi), we introduce another method of specifying the elements of a set: we use a criterion for membership in the set. With (vii)-(x), we give some examples of sets with non-mathematical elements. In (vii) and (viii), we again have small finite sets where we can list all the elements of the sets, whereas (ix) generalizes these to a sequence of arbitrarily large finite sets. In (ix) we take a leap of abstraction and pass from the sequence S n to an infinite set S. These sets, the S n and S, are widely used in mathematical finance we will discuss them again later in this lecture. 1 Actually, one of the first exercises in a more advanced course in set theory is to construct the real numbers as a certain set. In fact, the real numbers are constructed in terms of sequences of rational numbers. But that comes after the rational numbers are constructed in terms of the integers. And the integers are constructed in terms of the empty set and the operation of taking unions!
Exercise: Write an expression which denotes the set of points in R 2 which lie on the unit circle. Exercise: How many elements does S n have? Empty set There is a unique set that has no members at all. Quite naturally, it is called the empty set. The notation for the empty set is. Inclusion/Containment Let s take another look at examples (i)-(iv) above. There is an important relation between certain pairs of these sets that we want to focus on. Note that every element of the set 1,2, 3, 4,5 1,2,3,...,999. Similarly, every element of 1,2,3,...,999 Ν = { 1,2,3,... }. is also an element of is also an element of In such case, where every element of a set A is also an element of another set B, we say that A is included in B (or B contains A ), and we write A B. We also say that A is a subset of B. We formalize this definition as follows: Definition (inclusion/containment): A B if and only if x x A x B ( ) Exercise: Identify the inclusion relations that hold among the sets (i)-(x). Note that A and A A for every set A. Definition (power set): The set of all subsets of a set A is called the power set of A, and is denoted by Ρ A ( ).
(). Exercise: List all subsets of { 1,2, 3}, i.e., all elements of Ρ 1,2, 3 Exercise: If A is a set with n elements, how many elements does Ρ( A) contain? Equality of Sets We mentioned above that two sets are equal if they have exactly the same elements. One way of formalizing this is as follows: Definition (equality): A = B if and only if ( A B & B A) Sample Spaces and Events We use the notions introduced above to define the basic concepts of probability theory. Definition (sample space): The set of all possible outcomes of a particular experiment B is called the sample space of the experiment. Examples: Let the experiment B 1 consist of tossing a coin a single time. Then the sample space is S 1 = H,T, where H represents an outcome of heads and T represents an outcome of tails. Let the experiment B 2 consist of tossing a coin twice in a row. Then the sample space is S 2 = { HH,HT,TH,TT}. Let the experiment B consist of tossing a coin an infinite number of times. Then the sample space is S = { ω = ω 1 ω 2 ω 3 K : each ω i is a H or a T}. Not surprisingly, these samples spaces are often called coin toss spaces. See, for example, the early chapters of Shreve s Stochastic Calculus for Finance texts. Definition (event): An event is any collection of outcomes of a given experiment, that is, any subset of the sample space S. Note that the entire sample space S is itself an event. (Why?) Let A be an event, i.e., a subset of the sample space S. We say that the event A occurs if the outcome of the experiment is in the set A.
Operations on sets: unions, intersections, complements The fundamental set-theoretic operations are union and intersection. The intersection of two sets A and B, denoted A B, is the set of all elements which are elements of both A and B. The union of two sets A and B, denoted A B, is the set of all elements which are elements of A or B. We will go through some basic algebraic laws of the union and intersection operations. The first of these laws are that these operations are commutative and associative; we also note how they distribute over one another. Theorem: For any sets A,B,C, the following equations hold: (commutative laws) A B = B A A B = B A (associative laws) A B C A B C (distribution laws) A B C A B C ( ) = A B ( ) = A B ( ) = A B ( ) = A B ( ) C ( ) C ( ) A C ( ) A C ( ) ( ) Let us list some more quite basic properties of union and intersection: Theorem: For any sets A and B, A B = A if and only if A B A B = A if and only if B A A = A = A Proof: Exercise. It can be helpful to use the pictorial representations of set-theoretic operations, which are called Venn diagrams.
An additional operation on sets is complementation. The complement of a set A is the set of all elements which are not in A. When using complementation, we usually assume that everything is taking place inside a large fixed set, which is sometimes called the universe. For our purposes, we can take the universe to be a sample space S. Some basic properties of complementation: Theorem: For any sets A and B, which are subsets of a universe (sample space) S, ( A C ) C = A A A C = A A C = S A B A B ( ) C = A C B C ( ) C = A C B C The last two clauses of the theorem above are called DeMorgan s laws. They come quite often in probability and stochastic calculus, so you should get familiar with them. Recap (containment) A B iff x( x A x B) ( ) (equality) A = B iff A B & B A (union) A B = { x : x A or x B} (intersection) A B = x : x A and x B (complement) A C = x : x A Operations on sets: unions, intersections, complements The operations of union and intersection can be extended to infinite collections of sets. Such infinite unions and intersections also come up quite often in probability. If A 1, A 2, A 3,K is an infinite collection of sets, then U A i = { x : x A i for some i}
I = x: x A i for all i A i Example: Let S = (0,1] and A i = [(1/i), 1]. Then: U A i = U [(1/i),1]={ x [0,1] : x [(1/i),1] for some i}= (0,1] I A i = [(1/i),1] = I { x [ 0,1]: x [(1/i),1] for all i}= 1 {} Note: In probability and statistics, we usually only deal with countably infinite collections.
Definition (disjoint): Two events are disjoint (or mutually exclusive) if A B =. The events A 1, A 2, A 3,K are pairwise disjoint (or mutually disjoint) if A i A j = for all i j. Example: A i = [ i,i +1) for i = 0,1,2,K are pairwise disjoint. Definition (partition): If A 1, A 2, A 3,K are pairwise disjoint and that the collection A 1, A 2, A 3,K forms a partition of S. U A i = S, then we say