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CBSE NOTES ⇒ Class 11th ⇒ Mathematics ⇒ 1. Sets

1. Sets

Introduction


SETS 


Set: A collection of well defined objects is called a set. 

- Objects of collection is called "element" or "member".

Properties: 

(i) All elements of a set should have special property. 

(ii) They all should be differ to each other or they do not repeat. 

(iii) They should be well defined. 

  • Name of any set is writen in capital letter of English Alphabets

        A, B, C, D ........................ X, Y, Z

  • Elements and members of a set is writen in small letter of English Alphabets or using number system within a brace (medium bracket) with using commas { } 

        Example : A = {a, b, c, d} Or B = {1, 2, 3, 4}

Standard symbols of some special sets:

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+ : the set of positive rational numbers, and
R+ : the set of positive real numbers.

Symbols and their meaning: 

(i) " ∈ "  (epsilon) : "belong to" 

If a is an element of a set A, we say that “ a belongs to A”

in symbolic form we write it as : a ∈ A 

(ii) " ∉ " : "not belong to" 

When any element which not belong to any given set then we use symbol ∉ : "not belong to" 

 if a is not an element of set A. we say that "a not belong to A"

in symbolic form we write it as : a ∉ A 

(iii) " ⇒ " : emplies 

In common language "" means "emply toward ..... also ......" 

(iv) " = " : Equal to

A = B, it means set A is equal to set B. 

(v) " ≠ " : Not equal to 

A ≠ B, It means set A not equal to set B. 

(vi) “⇔” is a symbol for two way implications, and is usually read as " if and only if ". 

(vii) " : " : colon Here in the set colon stands for "such that" 

Representation of sets: 

Every set is introduced by its elements. So for expressing a set before we have to express its elements. 

Now, Method to express the elements of a set. 

(1) Roster Or tabular form : 

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,

(i) First five natural numbers: 

A = {1, 2, 3, 4, 5}. 

(ii) First four letter of English Alphabets

B = {a, b, c, d}.

(iii) The set of all natural numbers which divide 42

C = {1, 2, 3, 6, 7, 14, 21, 42}.

(iv) The set of all vowels in the English alphabet.

D = {a, e, i, o, u}.

Note: 

In Roster Form no-element is generally repeated

Example: 

The set of letters forming the word 'ELEMENT'

E = {E, L, M, N, T} 

Here no-element has been repeated. 

(2) Set-builder form 

In set-bulder form, all the elements of a set possess a single common property
which is not possessed by any element outside the set.

Example: 

D = { a, e, i, o, u} 

All elements of set A have a common property e.i vowel in English alphabet. 

Then, in set-builder form:

D = { x : x is a vowel in english alphabet } 

This set will be read as :

" the set of all x such that x is a vowel in english alphabet. "

C = {y : y is a natural number which divides 42}

B = (z : z is a first four letter of english alphabet} 

 

Other examples 

(i) A = {1, 2, 9, 25}  Roster form

Set-builder form

A = {x : x = n2, where n ∈ N and n < 6} 

(ii) B = {P, R, I, N, C, A, L}

Set-builder form

B = {x : x is a letter of the word PRINCIPAL}

(iii) C = {1, 2, 3, 6, 9, 18}

Set-builder form 

C = { x : x is a positive integer and is a divisor of 18}

(iv) D = (3, 6, 9, 12}

Set-builder form 

D = {x : x = 3n where n ∈ N and 0 < n < 5} 

 

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Mathematics Class - 11th

NCERT Maths book for CBSE Students.

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Study Materials List:

CBSE NOTES ⇒ Class 11th ⇒ Mathematics
1. Sets
2. Relations and Functions
3. Trigonometric Functions
4. Principle of Mathematical Induction
5. Complex Number and Quadratic Equations
6. Linear Inequalities
7. Permutations and Combinations
8. Binomial Theorem
9. Sequences and Series
10. Straight Lines
11. Conic Sections
12. Introduction to Three Dimensional Geometry
13. Limits and Derivatives
14. Mathematical Reasoning
15. Statistics
16. Probability

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