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# Discrete Mathematics I MATH 125

Mason

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This 1 page Class Notes was uploaded by Michale Kuhlman on Monday September 28, 2015. The Class Notes belongs to MATH 125 at George Mason University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/215006/math-125-george-mason-university in Mathematics (M) at George Mason University.

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Date Created: 09/28/15

Supplementary Lecture NatesfarMath 1251 Alexei V Samsanavich 212007 Lecture 2 Sets Relations and Functions A set S is a collection of distinct objects x1 x2 called elements or members of the set This is written as follows S x1 x2 and the expression x1 6 S says that x1 is a member of S belongs to S The notion of a set does not allow for multiple instances repetitions of the same element in the set while the notion of a sequence an ordered collection does The notion of a set does not relate to the notion of continuity while at the same time it provides a basis for all concepts of continuous data structures which are not a topic here As far as the set theory per se is concerned the nature of elements is irrelevant all that matters is how many elements are and whether two elements x and y are one and the same element x y or not x at y For example a set may contain another set or even itself as its element A set can be empty denoted by Q nite or in nite The number of elements in a set S is called the cardinality of S written as lSl Examples of sets are natural numbers N integers Z rational numbers Q real numbers R complex numbers C A nite set can be de ned by an explicit list of all its elements for example 1 2 3 and 2 are sets However 1 2 2 3 is not a set because the list contains two instances of one and the same element Alternatively a set can be de ned by a formula that speci es its elements for example x x e R x gt 2 LetA and B be two sets They are equal A B iff they contain the same elements or both are empty A is a subset of B contained in B A g B iff every element of A is an element of B In this case B is a superset of A B Q A We say that A is a proper subset ofB A C B or equivalently B is a proper superset ofA B DA iffA g B andA at B Negations are expressed by slashing the corresponding symbols for example e The following is true for any sets A B AgA QgA ABlt gtAgBBgA The set of all subsets of A is called the power set of A denoted PA The unionA U B is the set of all elements that appear in any of the two sets The intersection A n B is the set of all elements that appear in both sets The set di quoterenceAB or AB is the set of those elements of A that are not in B An intuition into these notions is given by the Venn diagram page 44 in the textbook Finally symmetric set difference denoted A or EB is defined as A EB B A U B A n B Cartesian product direct product of sets A and B denoted in this textbook as A X B is de ned as the set of all ordered pairs in each of which the rst element belongs to A and the second element belongs to B These elements are called coordinates of the ordered pair Thus an ordered pair is a sequence of two elements An ordered ntuple is a sequence of n elements If X is a set thenXquot the nth power of X not to be confused with the power set of X is the set of all ordered ntuples overX over X here means composed from elements ofX

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