**Set Theory Class 11 CBSE Board by Radius JEE Coaching in Aliganj Lucknow.**

**Set Theory Class 11 CBSE Board by Radius JEE is given below.****Sets are used to define the concepts of relations and functions.****The study of geometry, sequences, probability, etc. requires the knowledge of sets.****The theory of sets was developed by German mathematician Georg Cantor (1845-1918)****We often speak of collections of objects of a particular kind, such as, a pack of cards, a crowd of people, a cricket team, etc.**

### Examples of Collection of Object

**(i) Odd natural numbers less than 12, i.e., 1, 3, 5, 7, 9, 11**

**(ii) The rivers of India**

**(iii) The vowels in the English alphabet, namely, a, e, i, o, u**

**(iv) Various kinds of Quadrilateral**

**(v) Prime factors of 45, namely, 3, and 5**

**(vi) The solution of the equation: x2 – 7x + 12 = 0, viz, 3 and 4.**

**We note that each of the above example is a well-defined collection of objects**

**A set is a well-defined collection of objects.**

**We give below a few more examples of sets used particularly in mathematics, viz.**

** N : the set of all natural numbers**

** Z : the set of all integers Q : the set of all rational numbers**

** R : the set of real numbers Z+ : the set of positive integers**

**Q+ : Set of positive rational numbers, and**

**R+ : the set of positive real numbers.**

**(i) Objects, elements and members of a set are same thing.**

**(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.**

**(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.**

**If c is an element of a set A, we say that “c belongs to A” the Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write c ∈ A. If ‘k’ is not an element of a set A, we write k ∉ A and read “k does not belong to A”. Thus, in the set V of vowels in the English alphabet, a ∈ V but b ∉ V. In the set P of prime factors of 45, 3 ∈ P but 15 ∉ P**

## Methods of Representation of a Set

**There are two methods of representing a set:**

** (i) Roster or tabular form**

** (ii) Set-builder form**

## (i) Roster or tabular form

**(A) In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }. For example, the set of all even positive integers less than 9 is described in roster form as {2, 4, 6, 8}. Some more examples of representing a set-in roster form are given below:**

**(1) The set of all natural numbers which divide 12 is {1, 2, 3, 6, 12}**

**(2) In roster form, the order in which the elements are listed is immaterial. Thus, the above set can also be represented as {1, 3, 2, 12, 6}. But Element must not be repeated.**

**(3) The set of all vowels in the English alphabet is {a, e, i, o, u}**

**(4) The set of all odd natural numbers less than 9. Therefore, X = {1, 3, 5, 7} **

**(5) Set of all letters in the word MATHEMATICS. Therefore, Z = {M, A, T, H, E, I, C, S} **

**(6) The set of even natural numbers is represented by {2, 4, 6, 8. . .}. The dots tell us that the list of even numbers continue indefinitely.**

**(7) In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.**

**For example, in the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this propert**

## (ii) Set-builder form

**(B) In set-builder form, all the elements of a set possess a single common property**

**which is not possessed by any element outside the set. For example, in the set**

**{ 2,4,6,8}, all the elements possess a common property, namely, each of them**

**is an even natural number less than 9, and no other number possess this property. Denoting**

**this set by A, we write**

**A = { x: x is an even natural number less than 9}**

**Z = { x: x is a natural number and 4 < x < 12} is read as “the set of all x such that**

*x *is a natural number and *x *lies between 4 and 12.

**Let set of all natural numbers which divide 12 is {1, 2, 3, 4, 6, 12}. Written in Roster form**

** (Tabular form)**

**A = {1, 2, 3, 4,6, 12}**

**We can write Set A in Set builder form as A = {x: x is a natural number which divide 12}**

**Example 1 Write the solution set of the equation x2 + x – 20 = 0 in roster form.**

**Solution The given equation can be written as**

**( x – 4) (x + 5) = 0, i. e., x = 4, – 5**

**Therefore, the solution set of the given equation can be written in roster form as {4, – 5}.**

**Example 2 Write the set A = {1, 4, 9, 16, 25, . . . } in set-builder form.**

**Solution We may write the set A as**

**Z = { x : x is the square of a natural number}**

**Alternatively, we can write**

**Z = { x : x = n2, where n ∈ N**

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