1 A 3 minute lecture on set theory settheory7818 pdf and tex September 1 2010 A set is a collection of things We use letters often uppercase to denote a set Examples of sets S fthe set of squirrels who live on the C U campusg A fall male students at C U Boulder who are math majors and who drive BMWsg f g are called braces Some symbols are reserved for particular sets denotes the empty set a set with no members my set of friends and denotes the universal set the set that includes everything or at least everything being considered 1 1 1 As statisticans we care about sets and set theory because Put simply more later we care about set theory because it is a foundation of probability theory Consider the outcome of an experiment Then consider all of the possible outcomes the set of outcomes A particular outcome is an element in the set of possible outcomes We are often interested in the likelihood probability that an outcome will have some property e g the patient dies interest rates go up 10 cigarettes are consumed For example a treatment might characterized in terms of two random variables patient dies or does not patient turns green or not We might be interested in the likelihood that someone who gets the treatment dies All the oucomes with the property patient dies are a subset of all the possible outcomes of the experiment We want to know the likelihood that the outcome will belong to that subset 2 1 2 Some notation about the relationship between two sets X Y is the union of the sets X and Y that is the set that includes all the elements of X and Y X Y is the intersection of the X and Y that is those elements that belong to both X and Y Alternative notation is XY and X and Y XnY is the elements that belong to X but not to Y An alternative notation is X Y called a set di erence X nX X X is called the compliment of X Sometimes you will see the notation X c to denote the compliment of X So X X and X X Some additional notation and concepts x 2 A means x is an element in the set A So for example A B fx x 2 A and x 2 Bg M K means the set M is a subset of the set K 1 Sets X and Y are said to be equivelant if X Y and Y X 1 Sometimes we distinguish between subsets and strict subsets using to denote subset and to denote a strict subset With this more precise notation A B mean that A is a subset of B but there are elements in B that are not in A so A is a strict subset of B Whereas A B allows for the possibility that A B Note that A B and B In the notes we are using A A B to mean or 3 1 2 1 Some examples You can nd the union of two or more nite sets with the Mathematica command Evaluate by using the symbol between the sets I Evaluate f1 2 3g fa b cg f1 2 3 a b cg f1 2 3g f3 5g f7g f1 2 3 5 7g p p 2 3 9 r fa b cg r a b c 3 9 2 Now nd some intersections I Evaluate f1 2 3g f2 4 6g f2g fa b c dg fd e f g fdg f1 2 3g fa b cg f1 2 3g fg Note that and are algebraic commands but that they apply to sets rather than to variables Think of algebra for sets set algebra If two sets have no elements in common their intersection is the empty set denoted by empty brackets fg or the symbol You can evaluate combinations of union and intersection I Evaluate f1 2 3 cg f2 4 6g fa b cg f2 cg f1 2 3 cg f2 4 6g f1 2 3 cg fa b cg f2 cg 4 1 3 Venn diagrams What are Venn diagrams They are a way to represent sets and the relationships between those sets Sometimes the shapes and dimensions are important but often not Pictorially respresent the following using Venn Diagrams what is the universal set in each of these cases X Y X Y Z X Y Z Thanks to Anthony for the diagrams need to match up the names A is X and B is Y 5 Are the shapes of these Venn diagrams meaningful Is the following formula true XnY Y nX Demonstrate your answer with a Venn diagram 6 Now consider the following sets all hompsapiens who have ever lived all male homosapiens who ever lived all humans currently alive and all humans ever named Edward You can assume that females are never named Edward and that eveyone is either a male or female Draw a Venn diagram that indicates that most humans with the name Ed are dead 7 All humans who have ever lived males Named Ed alive I have assumed everyone was or is either a male or female not quite true and that half the births are female Those alive are a strict subset of those who have lived I drew the Ed set so that most of em are in the dead male category 8 females Now consider the following bad joke about deduction Assume three economists An econ prof at CU An undergraduate econ major at CU and an econ prof at MIT They are out hiking come to a ridge and look down into a valley They look down into the valley and see a bunch of cows all of which look black The prof at CU concludes on the basis of this observation that all cows are black The undergraduate say no then goes on to say that the observation proves only that all cows in this valley are black The MIT prof then asserts that what has been proven is that all cows in this valley are black on at least one side Using Venn diagrams represent the cow issue Draw your sets so that the econ prof at MIT is the only one correct Now draw the sets so that all three of the academics are correct 9 All cows Cows black on at least one side Black cows Cows in valley The white area is all cows that are not black on at least one side 2 The only thing that can be concluded from the information given in the joke is that cows in the valley are black on at least one side The above is an example of this but note that this diagram which is consistent with the information given is inconsistent with all cows being black or even all cows in the valley being black 2 Note here that the universal set is not everything it is all cows 10 All cows Cows black on at least one side Cows in valley …