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

## 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:-

### 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.

### 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.

### 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