Relations and Functions free pdf and video

 Relations and functions

NCERT Solutions  for Maths - Relations and Functions  by Sonika Anand Academy

Multiple Choice Questions

Video Lectures

Types of functions

Domain and Range of a function

How to find cartesian product of two sets A×B

Representation of two sets using arrow diagram

Find a relation between two sets where one set is cube root of other

How to find number of relations from one site to another

How to find number of elements in relation from a to b


Topic covered in given lecture are

Composition of a function 

How to find whether a  given function is function or not
How to calculate the value of function if given function is in algebraic form or trignometric form

How to add two functions
How to subtract two functions
How to multiply and divide two functions

How we can calculate value of fof and gof


Topic explained 
Find value of f(2),
Or f(x)etc 




Topics
Addition of two functions
Substraction of two functions
How to multiply two functions
Division of two functions




Functions are given in ordered form then how will you add subtract multiply divide the two functions and what is the range and domain of the resulting functions as they are given in the order form so must watch the video attached below




Cartesian product 

What is difference between function and Relation
How we find ordered pairs from two sets using cartesian product 
How ordered pair of Relations can be obtained from Cartesian product of two sets
If one set  A has m elements and another set B  has n elements then how will you find the number of functions obtained from two sets A and B = m×n


Content of Lecture

How to find domain and range and codomain of any function

How we will distinguish between one one and many one function

It is also explained as what is the other name of one one function that is one one function is also named as injective function whereas many one function is also known as surjective function

A  function as both one one and onto then it is also known as bijective function

 what are the different conditions which are satisfied when any function is one one function and when it is many one function

How meni define whether a function is strictly increasing or decreasing using graph method

How to find the number of relations in between two given sets that is what is the formula to find the number of relations between two sets

What is the formula to find the number of mappings from one set to another

Types of Functions



Inverse of a function 

What is an invertible function and how to find inverse of any function os explained in given lecture 





Binary Composition
This concept is  started in class 12 that is what are the commutative and associative properties of the binary functions

a*b=b*a. Commutative property
a*(b*c)=(a*b)*c associative property

How to check whether a function is infimum function or superinfimum function
How will we obtain a binary composition table for any given function



Domain and  Range of a Function 
Domain and range of a function is very important topic and generally asked in examinations but question is always asked there in IIT for internal exams of class 12th level so how to calculate to main and range of a function should be there in a there are one or more videos which I have attached below I must watch them properly and I think the concept of domain and range of a function header with the help of a diagram for with the help of order you can easily calculate how to find the domain and range


DOMAIN AND RANGE


Domain of Trignometric functions
In order to calculate the domain of trigonometric functions the concept is little bit different
So I made a separate video and audit explain how to find the domain and range for the trigonometric functions



Domain of modulus function
How to find domain of any modulus function ie |x|
Que discussed in attached lecture are --

Que-- Find domain and range for  f(x)= |x-3|
Que -- f(x) = (|x|-x)/2x
Que f(x) =( |x-4|)/x-4


Equivalence Relation
A function is an equivalence relation if a function  satisfies the three conditions
1 the function must be reflexive
2 a function must be symmetric
3 function  is transitive
Now what is reflexive and what are symmetric and what is transitive function and how will be show
All these concepts are explained in the lecture given below along with the questions

How to find a relation is equivalence or not 
For a equivalence relation equation is 
a) reflexive 
{(a,a)(b,b)}
b) symmetric 
{(a,b),(b,a)}
c) transitive 
{(a,b)(b,c),(a,c)}

Problem discussed 
If X= { 1,2,4...7}
R={(x,y)|(x-y) is divisible by 3} prove it is equivalence relation or not 

How to find Digraph of a Relation 


INJECTIVE or Surjective or Bijective 
How to find INVERSE of a function.              
If a function is oneone(injective) then
f(x)=f(x')
x=x' 
If a function is onto (surjective) then
 range = codomain 
Is a function is both one one and onto then that function is termed as bijective function
Problem 
Que --If f:N---N given by f(x) = 5x then find function is injective surjective or bijective

 Que --If f:A---B f(x) = (x-2)/(x-3) then show weather it is bijective or not Also find inverse 
Que -- f:R--[4, ♾️) f(x) = x²+4 show f is invertible find also its inverse




Questions discussed in next lecture --

How to find 

subset ,proper subset

 cardinal number of a set 

Subset =2^n

Proper subset = 2^n-1

Find number of subset 

If A={x:x = 3x+1,2<=x<=5}

Set is x= {2,3,4,5}

A={7,10,13,16}

So subset =2^n = 2⁴= 2×2×2×2= 16

How to find number of Relations from A to B containing m and n elements 

(2)^m×n

If two finite  sets A and B  have m and n  elements total number of relations is 64 find the value of m and n

If two sets have m and n elements and total number of subsets of first set is 56 more than subset of B  find the value of m and n



Summary - if A={1,2,3...m}. B ={1,2,3,....n}
Subset of set A - 2^m
Subset of B -2^n
Proper subset of A 2^m - 1



Set Theory

Problems discussed in given lecture--
Relations class 11 ex 2.1 

1)  If (x/3+1,y-2/3)= (5/3,1/3) find value of x and y 
2) If A set has 3 elements and the set
 B ={3,4,5} find the number of elements in A×B 

3) If G ={7,8} and H= {5,4,2}
Find G×H and H×G 

4) if A = {-1,1} find A×A×A

5) If A×B = {(a,x),(a,y),(b,x),(b,y)}
Find A and B 

6) If A ={1,2} B={ 3,4} find A×B and how many subsets A×B have 






How to prove DeMorgon's Law --

1) AUB = BUA 
U means union U means or 
2) AπB=BπA 
3) A-(BUC)= (A-B) π(A-C)
4) (AUB)'=A'πB'
5) A-B= AπB'


Problems  discussed in given lecture are --

1) --If X and Y are two sets that x has 40 elements XUY has 60 elements and X intersection  Y has 10 elements how many elements Y has?

2)--A college awarded 38 medals in football 15 in basketball 20 in cricket there medal  went to total of 58 men and only three men got medal in all three source how many received medals in exactly two or three sports



Multiple choice questions


1) The relation is defined on the set

A= {1,2,3,4,5} by R={(a,b):|a^2-b^2|<16} is given by

a) {(1,1),(2,1),(3,1),(4,1),(2,3)}

b) {(2,2),(3,2),(4,2),(2,4)}

c){(3,3),(4,3),(5,4),(3,4)}

d) none of these

Ans 

d )none of these


2) The smallest equivalence relation on the set A= {1,2,3}is

a) {(1,1),(2,2),(3,3)}

b){(1,1)}

c){(1,1),(2,1),(3,1),(2,2),(2,3),(3,2)}

d) none of these

Ans 

a) {(1,1),(2,2),(3,3)}


3) If A = {1,2,3}and B ={1,4,6,9} and R is a relation from Ato  B defined  X is greater than y the range of R is 

a) {1,4,6,9}

b){4,6,9}

c) {1}

d) none of these

Ans

c) {1}


4) A relation R is define from { 2,3,4,5 }  {3,6,7 ,10} by x R y = x is relatively prime to y then domain of R is 

a) {2,3,5}

b.){3,5}

c ) {2,3,4}

d) {2,3,4,5}

Ans 

a) {2,3,5}


5) R is a realation from {11,12,13} to {8,10,12} defined by y=x-3 Then inverse of R is

a) {(8,11),(10,13)}

b.) {(11,8),(13,10)}

c) {(10,13),(8,11),(8,10)}

d) none of these


Ans

a) {(8,11),(10,13)}


6) Let R be a relation on the set N of natural numbers defined by n R m iff n divides m Then R is 

a) Reflexive and symmetric 

b) Transitive and symmetric 

c) Equivalence

 d) Reflexive,transitive,but not symmetric

Ans 

d)

7) Maximum number of equivalence relations on the set A={1,2,3} is

a) 1

b). 2

c) 3

d) 5

Ans 

d ) 5


8) If the set A contains 7 elements and set B  contains 10 elements then number of one one functions from A to B is


Ans  b)

9) If A = {1,2,3,.....n} and B = {a,b}.Then number of subjections from A to B is 



Ans   (b)

10) If f(x)= px/x+1,xis not equal to -1 then for what values of p f(f(x))= x

a) √2

b) -√2

c) 1

d) -1

Ans 

d ) -1

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