chapter title Quick on set theory A set is a collection of things I will use letters often uppercase to denote a set Three examples are S fall squirrels who live on the University of Colorado campusg A fall male students at C U Boulder who are math majors with BMWsg and tennis sets in terms of who won each game each element is a sequence of at least six games T between two individuals A and B with the last two won by the same person 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 Every squirrel on campus would be an element in member of the S set In the T set if A indicates that individual A won game the element AAAAAA is the element outcome that individual A won the set by winning the rst six games and ABABABABABABBB is an element indicating that A and B traded wins for six games then B won the next two games to win the set 8 to 6 1 1 As statisticians 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 within a year of treatment or does not and patient loses at least ten pounds or not We might be interested in the likelihood that someone who gets the treatment lives and loses weight All the outcomes with the properties patient dies and patient losese at least ten pounds 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 In economic theories constraints are often assumed and these are sets constraint sets For example conside the consumer s budget set in consumer theory and the input requirement set in production theory 1 Think about all the di erent possible elements in the set of possible tennis sets Are the number of elements nite Countable Does the number of elements in this set depend on whether the match has the tie breaker rule 1 3 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 2 Sets X and Y are said to be equivalent if X 3 1 Y and Y X Some examples Unions 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 Intersections 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 2 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 A A B In these notes we are using to mean or 2 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 Combinations of union and intersections 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 Venn diagrams Venn diagrams attributed to John Venn circa 1880 are a way to pictorially represent sets and the relationships between those sets WolframMathWorld de nes them as A schematic diagram used in logic theory to depict collections of sets and represent their relationships http mathworld wolfram com VennDiagram html The universal set is often represented with a rectangle in two dimentional space the rectangle represents all possibilities but sometimes is simply implicit The dimensions of the rectangle need not have cardinal or ordinal meaning but can The objective of the Venn diagram is typically to provide a visual representation of the relationships between two or more sets in terms of the set properties and n and The sets to be considered e g X Y and Z are each represented an area in the rectangle 3 For example one might represent set X a strict subset of with 3 A Venn diagram is constructed with a collection of simple closed curves drawn in the plane According to Cyndi Joyce Aguzar 1918 the principle of these diagrams is that classes or sets be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram That is the diagram initially leaves room for any possible relation of the classes and the actual or given relation can then be speci ed by indicating that some particular region is null or is not null Venn diagrams normally comprise overlapping circles The interior of the circle symbolically represents the elements of the set while the exterior represents elements which are not members of the set Shapes other than circles can be employed and this is necessary for more than three sets Venn diagrams do not generally contain information on the relative or absolute sizes cardinality of sets i e they are schematic diagrams http en wikipedia org wiki Venn diagram To be precise most of the diagrams presented in this section are Euler diagrams not Venn diagrams but they are commonly called Venn …