__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’

** **__Types of Set:__

__Types of Set:__

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

__Finite Set:- __

**finite set**.

__Infinite Set:-__

**infinite Set.**

__Singleton Set:-__

**singleton set.**

__Null Set:-__

**null set.**

**empty**or

**void set.**

**Φ.**

__Subset:-__

__Subset:-__

__Super Set:-__

__Proper Subset:-__

__Equal Set:-__

__I__f 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.

__Universal Set:-__

__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 }

__Let A and B be two sets , then the__

**2. INTERSECTION:-****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**

^{c }

^{
}

^{
}^{
}^{
}^{
}^{
}^{ }

__ 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.

__Let A and B be two sets . The__

**5. SYMMETRIC DIFFERENCE OF SETS:**–**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)-(A∩B)

### __MULTISET:-__

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

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

- 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):a∈A, 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**R−1= {(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:- __

**x**

**∈**

**X**a unique element

**y**

**∈**

**Y.**

**X to Y**we write

**f:X**→

**Y.**

### 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:y**∈**Y and y = f(x) for all x**∈**X} . 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:-__

__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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