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    Add as FriendSet Theory Basic

    by: merry

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    1 : SET: For example ,suppose one defines the term set as a well defined collection of objects. Collection is a aggregate of objects or things. Aggregate is class of things. Class is collection. Their must be some undefined or primitive terms. In this Chapter we start with two undefined terms. Element and Set Assume that the word “ Set “ is synonyms with the words “Collection” , “aggregate” , “Class” and is comprised of elements. The words “Element” , “Object ” and “Member” are synonyms If a is an element of set A. Then write a ? A or a is in A or a is member of A
    2 : If a does not belongs to A then a ? A It is assumed here that if A is any set and a is any element then either a ? A or a ? A and possibilities are mutual exclusive. Thus , one cannot say “Consider the set A of some positive integers” . Because it is not sure whether 3 ? A or 3 ? A. The following are some illustration of sets : 1) The collection of Vowels in English alphabets ,This set contains five elements , Namely A , E , I ,O ,U. 2) The collection of first five prime natural numbers is a set containing 2,3,5,7,11 3) The collection of all status in the Indian union is a set. 4) The collection of past president of the Indian union is a set
    3 : 5) The collection of cricketers in the world who were out for 99 runs in a test match is a set. 6) The collection of good cricket player of India is not a set. Since the term “Good” player is vague and it is not well defined. Similarly , Collection of good teacher in a school is a set. However , the collection of all teachers in a school is a set. In this chapter we will have frequent interaction with some sets. N : Set of Natural Numbers. Z : Set of Integers. Z+: Set of all Positive Integers. Q : Set of all Rational Numbers. Q+ : Set of all Relational Positive Numbers. R : Set of all Real Numbers. R+ : Set of all Real Positive Numbers C : Set of all complex Numbers.
    4 : Description of a Set : A set is often described in the following two forms one can make use of any one of these two ways. According to convenience. Roster Form or Tabular Form . In this form a set is described by listing elements , separated by commas ,within braces {} . Example : The set of Vowels of English alphabets may be described as {a,e,i,o,u}. The set of even natural numbers can be described as {2,4,6,………}. Here the dots stand for and so on. If A is the set of all prime numbers less than A = {2,3,5,7} Note : The order in which the elements are written in a set makes no difference. Thus , {a,e,i,o,u} and {e,a,I,o,u} denote the same set. Also the repeataion of an element has no effect. For example {1,2,3,2} is the same as {1,2,3}
    5 : 2) Set – Builder Form : In this form a set is described by a characterizing property p(x) of its element x. In such a case the set is described by {x : p(x) holds} or , {x | p(x) holds} ,which is read as the set of all x such that p(x) holds ,the symbol ‘|’ or ‘:’ is read as such that. In other words , in order to describe a set , a variable x (say) is written inside the brace and then after putting a colon the common property p(x) possessed by each element of the set is written within the braces. Example : The set E of all even natural numbers can be written as E = {x : x is a natural number and x = 2n for n ? N} Or E = {x : x ? N , x = 2n , n ? N} Or E = {x ? N : x = 2n , n ? N}. The set of all real numbers greater then – 1 and less than 1 can be described as {x ? R : -1 < x > 1 } The set A = {1,2,3,4,5,6,7,8} can be written as A = {x ? N : x = 8}
    6 : Types of Set : 1 ) Empty Set : A set is said to be empty or null pr void set if it has no element and it is denoted by Ø . the roster method , Ø is denoted by {} . Example : {x ? R : x2 = -2 } = Ø {x ? N : 5 < x > 6 } = Ø The set a given by A = {x : x is an even number grater than 2 } is an empty set because 2 is the only even prime number. A set consisting of at least one element is called a non empty or non – void set. Note : If A and B are two empty set , then x ? A iff , x ? B is satisfied because there is no element x in either A or B to which the condition may be applied. Thus A = B. Hence there is only one empty set and we denote it by Ø . Therefore article ‘the’ is used before empty set.
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