Basics of Probability Notes - BASIC CONCEPTS OF SET THEORY...

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BASIC CONCEPTS OF SET THEORY --------------------- SETS The concept of a set and the properties of sets derived from certain fundamental assumptions or axioms form the basis of all modern mathematics. Axiomatic Set Theory derives the properties ot sets but does not define a set itself. The theory is based on an intuitive understanding of a set as a collection of entities or elements Notation: The standard notation for a set is as follows: The elements of the set under consideration are separated by commas and enclosed in curly brackets. For example a set X with elements Xl> X2, X3 and X4 is denoted as X= {Xl, x2, Xj, X4} Sets and their elements and defined terms are denoted in italics to differentiate them from other entities. Properties of Sets a. Set Membership Sets consist of elements or members. The elements of a set are its fundamental or atomic units A set may also be an element of another set For example, a line is set of points; the set of all lines defining a plane is a set of sets (ofpoints) All possible nesting of sets of sets are allowed as elements of other sets Notation: If X is an element of a set X, X is said to be contained in X and is denoted as follows: x E X b. Set Equality Two sets X and Y are said to equal if and only if they have the same elements. The equality of two sets, say X and Y is denoted as X = Yand their inequality by X = Y c. Subsets Most of set theory is aimed at creating new sets out of old ones. If X and Yare sets such that every element of X is also an element of Y, then X is said to be a subset of Yor that Y includes X This fact is denoted as follows: X c YorY :::> X Example: Let Y be a set of say a hundred individuals. we consider only those that are married and call the subcollection X, then X c Y The set X itself is denoted as: X = {x E Y . x is
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married j, where x is any element of Y The part enclosed by the brackets is to be read as "x in Y such that x is married" Some Important Properties of Subsets: i). According to the defintion of a subset any set X should include itself, i.e. X c X This property of set inclusion is said to be reflexive. ii.) If X, Y and Z are sets such that X c Y,Y c Z then X c Z . In this case set inclusion is said to be transitive iii). If Y c X and X c Y then X and Y have the same elements so that X = Y In fact the commonly used technique to prove the equality of two sets is to show that they include each other Note: i) The properties of belonging ( E ) and inclusion ( c) are conceptually very different Inclusion is always reflexive whereas belonging is not It is always true that a set X c X However, it would be difficult, if at all, to come up with an example of a set XEX ii) A set X is called countable if there exists a one-to-one correspondence between the set of all integers and the elements of X A set X isfinite if for some positive integer 'n' there exists a one-to-one correspondence between the elements of the set (I, 2, 3,. ..n) and the elements of X. For example, the rational numbers are countable, whereas the set of reals are uncountable d. Power Set: Theorem: If X = [x j' x z, x,J
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This note was uploaded on 10/27/2007 for the course MATH 4710 taught by Professor Jung during the Fall '05 term at Cornell University (Engineering School).

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Basics of Probability Notes - BASIC CONCEPTS OF SET THEORY...

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