What is sets - Definition | Meaning | Example | IIT JEE Math

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What is Sets (समुच्चय क्या है )

Object :- In Mathematical language all living and non-living things in universe are known as objects. 

(गणितीय भाषा में Object ब्रह्मांड में सभी सजीव और निर्जीव वस्तुएँ कहलाती हैं।)

Sets - The collection of well defined distinct objects is known as a set. 

Generally , Sets are denoted by capital letter A,B,C.... and its elements are denoted by small letters are denoted by small letters a,b,c

समुच्चय - सुपरिभाषित वस्तुओं के संग्रह को समुच्चय कहते हैं। आम तौर पर, सेट को बड़े अक्षरA,B,C ... द्वारा दर्शाया जाता है और इसके तत्वों को छोटे अक्षरों a,b,c.. से दर्शाया जाता है, 

Ex- A = { a , b, c, d } 

Here A =  denoted set 

         a,b,c,d = denoted by element of set 

Example - 

Let A is a non-empty set, if x is an element of A , then we write 'x∊A' 

and read as 'X is an element of A ' or  'X belongs to A' . 

if x is not and element of of A , 

then we write ' x ∉ A 

Read as x is not an element of A ' or x does not belongs to A ' 

Example 

A = Set of all vowels in English alphabets. 

In this set a , e , i , o and u are members .

"The Theory of Sets was developed by German Mathematician Georg Cantor ( 1845-1918). The concept of sets is widely used in the foundation of relations, functions , logic , probability theory , etc. According to cantor ' A set is any collection into a whole of definite and distinct objects of our intuition of thought , "

Some Example of sets are - 

(i) A set consisting of members of Lok Sabha 

(ii) the set consisting of all days in a week 

(iii) the set of solutions of the equation 

            x²-8x+15=0 viz 3 and 5

(iv) the set of prime numbers <30

(v) the set of  prime factors  of 210 viz , 2,3,5 and 7

(vi) The set of all positive integers 

        { 1,2,3,....n....}

(vii) the set of all even integers 

    

     {.....-4 , -2 , 0 , 2 , 4 , 6 .....}

(viii) The set of all integers divisible by 5 

Representation of a Set - 

we are use the following two methods to represent a set. 

(i) Roster ,Listing or Tabular Method - In this Method , elements are listed and put within a braces { } and separated by commas. 

Example - 

 

A set of first five integers. 

 

A = { 1,2,3,4,5}

A = set of all prime numbers less than 11 . 

A = ( 2,3,5,7}

A = set of all positive even integers. 

A = 2 ,4,6,8 

(ii) Set Builder , Rule or Property Method - In this Method , instead of listing all elements of a set, we list the property or properties or properties satisfied by the elements of set and write it as . 

A = { 1,2,3,4,5,6,7,8}

= { x : x∊ N and x ≤ 8 } 

Notation of Some standard sets - 

(i) Set of  all natural numbers N =  { 1,2,3....}

(ii) Set of all whole numbers , W =  { 0,1,2,3,....}

(iii) a. set of all integer " s, I or Z = { ......-2,-1,0,1,2,.....}

        b. set of all positive or negative integers . 

                

                 I⁺ = {1,2,3,........∞}

                 I^- = {-1,-2,-3,........∞}