Chapter 1 Set Theory ,Relation, Functions and Natural number / cbse class 11th

Chapter – 1 Notes

Cbse class 11th set notes

Set Theory , Functions and Natural Numbers

1. SET: 

  • A set is a collection of well defined objects ,called elements or members of the set.
  • These elements may be anything like nmbers ,letters of alphabets, points etc.
  • Sets are denoted by capital letters and theirs elements by lower case letters.
  • If an object  x is an element of set A, we write it as x∈A and read it as ‘x belongs to A’ other wise x does not belongs to A.

                      Types of Set:

  • Finite Set
  • Infinite Set
  • Singleton Set
  • Null Set
  • Subset
  • Superset
  • Proper  Subset
  • Equal Set
  • Universal Set
  • Disjoint Set

Finite Set:- 

A set which consist finite number of elements is called finite set.

Infinite  Set:-

A set which consist infinite number of elements is called infinite Set.

Singleton Set:-

A set which has consist only one element is called singleton set.

Null Set:-

A  set which contains no element at all is called null set.
It is also known as empty or void set.
It is denoted by {} or Φ.

Subset:-

Let A and B be two sets , if every elements of A belongs to B i.e., if every elements of set A is also an element of set B , then A is called subset of B andit is denoted by      A⊆B.

Super Set:-

If A is subset of a set B , then B is called superset of A. 

Proper Subset:-

Any subset A is said to be proper subset of another set B, if there is at least one element of B which does not belong to A, i.e, if A⊆B but A is not equal to B.
It is denoted by A⊂B.

Equal Set:-

If two sets A and B are said to be equal if every element of  A belong to set B and every element of B belong to set A.
It can be written as A=B.

Universal Set:-

In many applications of set , All the sets under consideration are considered as subsets of one particular set.
This set is called universal set .
It is denotedby U.

Disjoint Set:-

Let A and B be two sets , if there is no common element between A and B, then they are said to be Disjoint set.

Types of Operation on SETs:-

1. UNION:- Let A and B be two Sets, then the union of set A and B is a set that contains those element that are either in A or B or in both .It is denoted by AUB.

Symbolically, AUB = { x|x∈A  or x∈B }

2. INTERSECTION:- Let  A and B be two sets , then the intersection of A and B is a set that contain those element which are common to both A and B . It is denoted by A∩B  and is read as ‘A intersection B’. 






 3.COMPLEMENT:- Let U be the universal set and A be any subset of U, then complement of A is set containing elements of U which do not belong to A.  It is denoted by  A   








                                                                                                                                   
4. DIFFERENCE OF SETS:- Let A and B be two sets . Then difference of A and B is a set of all those elements which belong to A but do not belong to B and it is denoted by A-B.





5. SYMMETRIC DIFFERENCE OF SETS: Let A and B be two sets . The symmetric difference of  A and B is a set containing all the elements that belong to A or B but not both . It is denoted by A△B .                                                                                    Also, A△B = (AUB)-(AB)

                                                                                                                           MULTISET:-

  • Multisets are sets where an element appear more than once,

For ex:- A = { 2,2,2,3,3,4}

               B= {a,a,a,b,b,b,c,c}
are multisets.
  • These multisets A and B can also be written as                                                          A = {3.2,2.3,1.4} and B = {3.a,3.b, 2.c}
  • The multiplicity of an element in a multiset is defined to be number of times an element appears in the multiset.In above examples, multiplicities of the elements 2,3,4 in multiset A are 3,2,1 respectively.
  • Let A and B be two multisets. Then A union B , is the multiset where the multiplicity of an element is the maximum of its multiplicities in A and B.
  • The difference of A and B , A-B is the multiset where the multiplicity of an element is equal to multiplicity of the element in A minus the multiplicity of the element  in B if the difference is positive , and is equal to zero if the difference is zero and negative.
  • The intersection of A and B , A intersection B is the multiset where the multiplicity of an element is the minimum of its multiplicities in A and B.
  • The sum of A and B , A+B is the multiset where the multiplicity of an elements is the sum of multiplicities of the element in A and B.

 

Relation in set :  Let A and B be two non empty sets, then R is relation From A to B if R is subset of A x B and is set of ordered pair (a,b) where  a belongs A and b belongs to B . It is denoted by aRb and read as ” a is related to b by R”.

R= { (a,b):aA, b∈B, aRb}

Types of Relations:-

  • Identity Relation :- A relation R is called identity relation on A If R= {(a,a) | a∈A}.           It is also called diagonal relation.
  • Void Relation :- A relation R is called void relation  on A if R=Φ ( phi symbol ) . It is also called null relation.
  • Inverse Relation :- A relation R is defined from B to A is called inverse relation of R defined from A to B if                                                                                                                                                                                                                                               R1= {(b,a):b∈B and a∈A and (a,b)∈R}. ( Inverse relation symbolic form)
  • Complement of a Relation :- Let relation R is defined from A to B ,
  • then complement R is set of ordered pair {(a,b):(a,b)∉R} ( ∉ math symbol “does not belong” ). It is also called Complementary relation .
  • Universal Relation :- A relation R is called universal relation on A if R= A x A . In case R is defined from A to B, then R is universal relation if R=AxB.  

 Properties of Relation:-

  • Reflexive Relation
  • Irreflexive Relation
  • Non-reflexive Relation
  • Symmetric Relation
  • Asymmetric Relation
  • Antisymmetric Relation
  • Transitive Relation

                 Equivalence Relation :-
A relation R is said to be equivalence relation if it is Reflexive , Symmetric and Transitive.
For a equlvalence Relation satisfies following three properties:



Que: Let X = {1,2,3,4,5,6,7} and R = {(x,y)|(x-y) is divisibleby 3}. Is R Equivalence relation.
Solution:-

Function:-  

Let X and Y be two non-empty set . A function from X to Y isa rule that assign to each element  X a unique element Y.
If f is a function from X to Y we write f:XY. 
Function are denoted by f,g,h,i etc.

Domain and co-domain  of function:-    

Let f be a function from X to Y . Then set X is called domain of function f and Y is called co-domain function f.

Range of Function :-

The range of f is set of all images ofelement of X.                                                                     i.e., Range(f) = {y:yY and y = f(x) for all xX} . Range (f) ⊆ Y.

Classification of Functions: –

1.Algebraic function

  • Polynomial function
  • Rational Function
  • Irrational functions

2. Transcendental functions

  • Trigonometric function
  • Inverse Trigonometric functions
  • Exponential funnctions
  • Logarithmic Functions

Operations / types on function:-

1. One to One Funtion (Injective Function or injection)
Let f:X tends to Y then f is called one-to-one function if for distinct element of X there are distinct image in Y i.e., f is one-to-one if 
f(x1)=f(x2) implies x1=x2 for all x1,x2, belongs to X

 

2. Onto Function (Surjection or Surjective function)
Let f:X tends to Y then f is called onto function iff for every element y belongs Y there is an element x belongs to X with f(x)=y or f is onto if Range (f)=Y.

  
3. One-to-One Onto function (Bijective or Bijection)
A function which is both one-to-one and onto is called one-to-one onto function.

 

4. Many One function
A function which is not one-to-one is called many one function i.e., two or more elements in domain have same image in co-domain i.e.,
If f:X tends to Y then f(x1)=f(x2) implies x1 not equal to x2.

5. Identity Function
6. Inversse Function

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1 Response

  1. Rtsall says:

    Related to this topic any kind of problem you face it , you can easily ask in the comment section.
    Thankyou

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