CLASS-11 SETS FULLY Concept ( बहुत ही आसान भाषा में)

         

                       TOTAL TOPYC OF SETS                       

 1. What is sets.( set किसे कहा जाता है)

2. Notation.( Set के प्रतीक)

3. Some Standard Notation(कुछ महत्वपूर्ण प्रतीक)

4. Representation (set प्रदर्शित करना)

5. Types of Sets (set के प्रकार)

6.Sub-Set of Set (समुच्चय के उप-समुच्चय)

7. Interwals( अंतराल)

8. Operation of sets(समुच्चय संक्रिया)

9.Algebra of sets ( बीजगणितीय प्रयोग)

10. Venn Diagram ( वेन आरेख)

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अब हम एक-एक टॉपिक के बारे में विस्तार से अध्ययन करेंगे

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1 . SETS- 

A well defined collection of  distinct objects/ elements/ numbers/ members / persons  is called set.

(Well defined=which collection not change person to person)

(सभी व्यक्तियों द्वारा किया गया कलेक्शन एक जैसा होना चाहिए उसे ही well defined कहते हैं)

Ex. Collection of odd natural Numbers less than 10.

Ans  Sonu: 1,3,5,7,9.          Monu:. 1,3,5,7,9

Well defined -. It is a set.

Ex. Collection of 3 odd Numbers less than 10.

Sonu: 1,3,7.             Monu: 1,5,7

Not well defined- It is not a set.

Ex. Collection of first 3 Prime minister of India.

Sonu: Jawahar Lal Nehru, Gulzai Lal Nanda,Lal Bahadur Shastri

Monu: Jawahar Lal Nehru, Gulzai Lal Nanda,Lal Bahadur Shastri

Well defined- It is a set.

Ex: Collection of 3 prime minister of India.

Sonu:  Narendra Modi, Charan Singh, Indra Gandhi

Monu: Rajiv Gandhi, Narendra Modi, Indra Gandhi

Not well defined- It is not a set

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2.  NOTATION:.  

(I) SET-. Capital latters A,B,C,...X,Y,Z..etc

(II) ELEMENTS- Small letters a,b,c..x,y,z..etc

EX: Collection of odd natural Numbers less than 10.

Ans: 1,3,5,7,9   

(set form- use commas and { } bracket. 

A ={1,3,5,7,9 }

If 1 is a element of set A = 1 belongs to A                                                      = 1∈A

If 8 is not a element of set A= 8 not belong to A

8∉ A 

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3. SOME STANDARD NOTATION TO REPRESENT SET

(I). Natural Numbers.  N ={1,2,3,...}

(II). Whole Numbers.     W= {0,1,2,3...}

(III). Integers.     I or Z ={...,-3,-2,-1,0,1,2,3...}

(IV). Real Numbers.  R

(V). Even Numbers.  E= {2,4,6,...}

(VI). ODD Numbers.   O= {1,3,5,...}

(VII). Rational Numbers. Q ={ P/Q. Q≠0}

(VIII) Irrational Numbers .T ={√2,√3,√5,π...}

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4. REPRESENTATION OF SET:

(I). Statement Form

 example, the set of even numbers less than 15.

In statement form, it can be written as 

{even numbers less than 15}.

(II). Roster Form

 Example: the set of natural numbers less than 5.

Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,……….

Natural Number less than 5 = 1, 2, 3, 4

Therefore, the set is N = { 1, 2, 3, 4 }

(III).Set Builder Form

Example: Write the following sets in set builder form: A={2, 4, 6, 8}

 the set builder form is A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}

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5. TYPE OF SETS:

(I). Empty/Null/Void Set:  

A set which does not contains any element.

Denoted by.  {  }.

Ex. { X: X²= 4, X is a odd number}

       {X: X² = -1, X ∈ R

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(II) Singleton/ Unit Set: A set which contains only one element.

Ex.    { 0 },. { 13 }.   etc

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(III). Finite Set:  A set which contains finite numbers of elements.

Ex. A ={ 1,2,5,7 },.    B = { x : x² = 25, X∈ R }

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(IV). Infinite Set:  A set which contains infinite numbers of elements.

Ex: A={1,2,3,4,5,.....}

Others Example of infinite sets........

Natural Numbers, 

prime numbers, 

even Numbers

Odd Numbers

Integers

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(V). Equal Set: Two  sets which contains same elements.

If  A= {1,2,3} ,B ={1,3,2}. Then. A=B

If  A= {1,2,3 } ,B ={a,b,c}. Then. A # B

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(VI). Equivalent Set: Two sets which contains same  numbers of elements.

The order of sets does not matter here. It is represented as:

 n(A) = n(B)

If A= {1,2,3 } ,B ={a,b,c}. Then. A ~ B

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(VII).Disjoint Set: Two sets which does not contains common elements.

  A= {1,2,3} ,B ={ 5,6,7 }. Then. A∩B =Ø

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(VIII). Overlapping Set:  Two sets which contains at least one common element.

A= {1,2,3} ,B ={ 5,2,7 }. Then. A∩B ={ 2}

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(IX). Family  of Set: A set contains elements which are it self sets.

Ex.  A= { 1,{2,3},5,{6}, 8 }

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(X). Universal Set : A set which contains all elements of all sets.

Denoted by U.

Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be:

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

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6. SUBSET OF A SET:

If every element of set A is also an element of another set B, then A is called subset of B.

Denoted by  A ⊆ B . 

A= {1,2,3,4}

B={ 1,3}

C={1,4}

 

Set B is a part of set A 

= B is subset of A 

=  B  ⊆A,.   

Set C is a part of set A 

= C is subset of A 

= C ⊆  A   

Numbers of subset of a Set = (2)ⁿ

If A = {1,2},

 then.   Possible Sub set ={1}, {1,2}, Ø

Subset~ part of set

Proper Subset

If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B.

Example: If A = {2,5,7} is a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7}

But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.

N  ⊂  W   ⊂  Z  ⊂ Q ⊂     R

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Superset

Set A is said to be the superset of B if all the elements of set B are the elements of set A. It is represented as A ⊃ B.

For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of B.

Power Set. : The collection of all subsets of a set or set of subset. denoted by P(A).

Ex. A= { 1,2 } Then p(A)={ Ø,{1},{2},{1,2} }

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7. INTERVALS: 

Intervals are the subset of real numbers which contains elements lying between two specific real numbers.

(I). Closed Intervals. [  ]  :  

{x : x ∈ R, a ≤ x ≤ b}  =   [ a , b ].

(II) Open Intervals.   (  ).  :  

{x : x ∈ R, a < x< b }  =   (a , b ).

(III) Semi Closed open intervals.    [   ). :  

{x : x ∈ R, a ≤ x <  b }  =   [ a , b ).

(IV) Semi open Closed intervals. (   ]. :  

{x : x ∈ R, a < x ≤ b }  =   (a , b ]. 

{x : -∞ ≤  x ≤  ∞} = (-∞, ∞).

{x :  -∞ < x  < ∞} = (-∞, ∞).

Intervals	Notations
Empty interval	[b, a] = (b, a) = [b, a) = (b, a] = (a, a) = [a, a) = (a, a] = { } = Φ
Degenerate interval	[a, a] = {a}
Open interval	(a, b) = {x : a < x < b}
Closed interval	[a, b] = {x : a ≤ x ≤ b}
Left-closed, right-open interval	[a, b) = {x : a ≤ x < b}
Left-open, right-closed interval	(a, b] = {x : a < x ≤ b}
Left-bounded, open and right-unbounded	(a, +∞) = {x : x > a}
Left-bounded, closed and right-unbounded	[a, +∞) = {x : x ≥ a}
Left-unbounded and right-bounded, open	(-∞, b) = {x : x < b}
Left-unbounded and right-bounded, closed	(-∞, b] = {x : x ≤ b}
Unbounded interval at both ends	(-∞, +∞) or [-∞, +∞] = R

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8. OPERATION ON SETS:

(I).Union of Sets

If set A and set B are two sets, then A union B is the set that contains all the elements of set A and set B. It is denoted as A ∪ B.

Example: Set A = {1,2,3} and B = {4,5,6}, then A union B is:

A ∪ B = {1,2,3,4,5,6}

(II). Intersection of Sets

If set A and set B are two sets, then A intersection B is the set that contains only the common elements between set A and set B. It is denoted as A ∩ B.

Example: Set A = {1,2,3} and B = {4,5,6}, then A intersection B is:

A ∩ B = { } or Ø

Since A and B do not have any elements in common, so their intersection will give null set.

(III). Difference of Sets

If set A and set B are two sets, then set A difference set B is a set which has elements of A but no elements of B. It is denoted as A – B.

Example: A = {1,2,3} and B = {2,3,4}

A – B = {1}

B - A    = {4}

Example : 

Find A U B and A ⋂ B and A – B.

If A = {a, b, c, d} and B = {c, d}.

Solution: 

A = {a, b, c, d} and B = {c, d}

A U B  = {a, b, c, d} 

A ⋂ B = {c, d} and 

A – B = {a, b}

(IV). Complement of Sets

The complement of any set, say A, is the set of all elements in the universal set that are not in set A. It is denoted by A’.

Ex:  U = { 1,2,3,4,5,6,7,8,9}. A= {4,5,6,7}

A' = {1,2,3 } 

(V). Cartesian Product of sets

If set A and set B are two sets then the cartesian product of set A and set B is a set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. It is denoted by A × B.

We can represent it in set-builder form, such as:

A × B = {(a, b) : a ∈ A and b ∈ B}

Example: set A = {1,2,3} and set B = {a, b}, then;

A × B = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)}

(9). ALGEBRA OF SETS.

For any three sets A, B and C

n ( A ∪ B ) = n(A) + n(B) – n ( A ∩ B)

If A ∩ B = ∅, then n ( A ∪ B ) = n(A) + n(B)

n( A – B) + n( A ∩ B ) = n(A)

n( B – A) + n( A ∩ B ) = n(B)

n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B )

n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) +  n ( A ∩ B  ∩ C)

Properties of Sets

Commutative Property : A∪B = B∪A A∩B = B∩A

Associative Property : A ∪ ( B ∪ C) = ( A ∪ B) ∪ C A ∩ ( B ∩ C) = ( A ∩ B) ∩ C

Distributive Property : A ∪ ( B  ∩ C) = ( A ∪ B)  ∩ (A ∪ C) A ∩ ( B ∪ C) = ( A ∩ B) ∪ ( A ∩ C)

De morgan’s Law : Law of union           : ( A ∪ B )’ = A’ ∩ B’ Law of intersection : ( A ∩ B )’ = A’ ∪ B’

Complement Law : A ∪ A’ = A’ ∪ A =U A ∩ A’ = ∅

Idempotent Law And Law of a null and universal set : For any finite set A A ∪ A = A A ∩ A = A ∅’ = U ∅ = U’

(10).VENN DIAGRAM: 

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Symbol	Symbol Name	Meaning	Example
{ }	set	a collection of elements	A = {1, 7, 9, 13, 15, 23}, B = {7, 13, 15, 21}
A ∪ B	union	Elements that belong to set A or set B	A ∪ B = {1, 7, 9, 13, 15, 21, 23}
A ∩ B	intersection	Elements that belong to both the sets, A and B	A ∩ B = {7, 13, 15 }
A ⊆ B	subset	subset has few or all elements equal to the set	{7, 15} ⊆ {7, 13, 15, 21}
A ⊄ B	not subset	left set is not a subset of right set	{1, 23} ⊄ B
A ⊂ B	proper subset / strict subset	subset has fewer elements than the set	{7, 13, 15} ⊂ {1, 7, 9, 13, 15, 23}
A ⊃ B	proper superset / strict superset	set A has more elements than set B	{1, 7, 9, 13, 15, 23} ⊃ {7, 13, 15, }
A ⊇ B	superset	set A has more elements or equal to the set B	{1, 7, 9, 13, 15, 23} ⊇ {7, 13, 15, 23}
Ø	empty set	Ø = { }	C = {Ø}
P (C)	power set	all subsets of C	C = {4,7}, P(C) = {{}, {4}, {7}, {4,7}} Given by 2s, s is number of elements in set C
A ⊅ B	not superset	set X is not a superset of set Y	{1, 2, 5} ⊅{1, 6}
A = B	equality	both sets have the same members	{7, 13,15} = {7, 13, 15}
A \ B or A-B	relative complement	objects that belong to A and not to B	{1, 9, 23}
Ac	complement	all the objects that do not belong to set A	We know, U = {1, 2, 7, 9, 13, 15, 21, 23, 28, 30} Ac = {2, 21, 28, 30}
A ∆ B	symmetric difference	objects that belong to A or B but not to their intersection	A ∆ B = {1, 9, 21, 23}
a ∈ B	element of	set membership	B = {7, 13, 15, 21}, 13 ∈ B
(a, b)	ordered pair	collection of 2 elements	(1, 2)
x ∉ A	not element of	no set membership	A = {1, 7, 8, 13, 15, 23}, 5 ∉ A
|B|	cardinality	the number of elements of set B	B = {7, 13, 15, 21}, |B|= 4
A × B	cartesian product	set of all ordered pairs from A and B	{3,5} × {7,8} = {(3,7), (3,8), (5,7), (5, 8)}
N1	natural numbers / whole numbers  set (without zero)	N1 = {1, 2, 3, 4, 5,…}	6 ∈ N1
N0	natural numbers / whole numbers  set (with zero)	N0 = {0, 1, 2, 3, 4,…}	0 ∈ N0
Q	rational numbers set	Q= {x | x=a/b, a, b ∈ Z}	2/6 ∈ Q
Z	integer numbers set	Z= {…-3, -2, -1, 0, 1, 2, 3,…}	-6 ∈ Z
C	complex numbers set	C= {z | z = a + bi, -∞<a<∞,                         -∞<b<∞}	6 + 2i ∈ C
R	real numbers set

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