Relation and Functions Class XII Chapter 1

Class 12  chapter 1

Relations and functions

Basic concepts of topic relations and functions class XII chapter 1 of mathematics. Equivalence relations, different types of functions, composition and inverse of functions.

Introduction

In class 11 we have studied about Cartesian product of two sets, relations, functions, domain, range and co-domains. Now in this chapter we have studied about the different types of relations, different types of functions, composition of functions and invertible functions.

Relations

A relation R from set A to set B is the subset of Cartesian product A x B.

In Cartesian product A x B number of relations equal to the number of subsets.

Domain

Set of all first elements of the ordered pairs in the relation R is called its domain.

Range

Set of all second elements of the ordered pairs in the relation R is called its range.

In any ordered pair second element is also called the image of first element.

Types of relations

i) Empty Relation:

A relation R in set A is called empty relation, if no element of A is related to any element of A, i.e., R = φ  ⊂ A x A

Example: Let A = {1, 2, 3} and R be a relation on the set A defined as R = {(a, b) : a + b = 10; a, b ∈ A}

Here R is empty relation. R = Φ

ii) Identity Relation

A relation R in set A is called identity relation, if every element of A is related to itself, i.e., R = {a, a)  ⊂ A x A

A relation R is an identity relation in set A if for all a ∈ A, (a, a) ∈ R.

iii) Universal Relation :

A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A x A

iv) Reflexive relation

A relation R in a set A is called reflexive, if (a, a) ∈ R,  for every a ∈ A

v) Symmetric Relation

A relation R in a set A is called symmetric, if (a, b) ∈R implies that (b, a) ∈ R, for all a, b ∈ A.

vi) Transitive Relation

A relation R in a set A is called transitive, if (a, b) ∈R and (b, c) ∈R, implies that (a, c) ∈ R, for all a, b, c ∈ A

vii) Equivalence relation:

A relation R in set A is said to be equivalence relation if it is reflexive, symmetric  and transitive relation.

Viii) Equivalence Relations:

Let R be an equivalence relation on a set A and let a ∈ A. Then we define equivalence class of a as 

[a] ={b ∈ A : b is related to a} = {b ∈ A : (b, a) ∈ R}

Note : Both empty relation and universal relations are sometimes called equivalence relation of trivial relation.

Functions 

One-One functions (or injective functions):

A function f : A →B is said to  be one –one or injective function if different elements of set A has different images in set B

Mathematically: 

Case 1: For all a, b ∈ A, we have: If  f(a) = f(b) ⇒ a =  b
Case 2 :  For a, b ∈ A, such that : If a ≠ b ⇒ f(a) ≠ f(b)

In the above figure different elements of set A have different images in set B. So this function is called one-one function.

Set A has 3 elements and the set B has 4 elements. Then the number of injective functions that can be defined from set A to set B is  = ⁴P₃ = 4! = 24.

Generalization

Set A has m elements and the set B has n elements (n > m). Then the number of injective functions that can be defined from set A to set B is  =  ⁿP_m  = n! 

If n < m then Then the number of injective functions that can be defined from set A to set B is = 0

If a function has m elements and is defined from A to A then number of injective relations from A to A = n!

Many One functions:

If two or more elements of set A have same image in set B then it is called many one function.

If a function is not one-one then it is called many one.

Here two different elements a and b of set A has same image in set B. So it is not an one-one function. It is only a many one function.

Onto function ( or surjective function)

A function f : A →B is said to be onto (or surjective) function, if every element of set B is the image of some element in set A.

Let y ∈ B, then for every  element  y ∈ B,  ∃ (there exist) an element   x ∈ A such that  f(x) = y

In the above figure every element of set B is the image of some element in set A so it is a onto function. It is neither  many one nor   one-one.

If n(A) = p, then number of bijective functions from set A to A are p!
Because If n(A) = p and n(A) = n(B) then the number of bijective functions from A to B is p!

Other definitions of onto functions are as follows

A function f : A →B is said to be onto (or surjective) function, if every element of set B has pre - image in set A.

If range of the function is equal to the co-domain(or set B), then it is called onto function.

Exercise 1.2

Q No. 9

\[f:N\rightarrow N,\; \; f(n)=\left\{\begin{matrix} \frac{n+1}{2}, if\; \; n\; \; is\; \; odd\\ \frac{n}{2}, if\; \; n \; \; is\; \; even \end{matrix}\right.\]

State whether the function f is bijective. Justify your answer.

One-One

\[Let\; (1,2)\epsilon N\; then\]\[f(1)=\frac{n+1}{2}=\frac{1+1}{2}=\frac{2}{2}=1\]\[f(2)=\frac{n}{2}=\frac{2}{2}=1\]\[f(1)=f(2)\: but\: \: 1\neq 2\]⇒ f is not one one function.

Onto

Case 1: If n = odd = 2q+1=4q+1\[f(n)=\frac{n+1}{2}\]\[f(4q+1)=\frac{4q+1+1}{2}=\frac{4q+2}{2}=2q+1\]⇒ 2q + 1 is also an odd function ∈ N

Case 2: If n = even = 2q = 4q, \[f(n)=\frac{n}{2}=\frac{4q}{2}=2q\]⇒ 2q  is also an even function ∈ N

From case 1 and case 2 we conclude that :

*For all odd numbers in the domain we have odd numbers in the co-domain .

*For all even numbers in the domain we have even numbers in the co-domain .

⇒ R_f = Odd natural number and Even natural numbers.

⇒ R_f = All Natural Numbers = Co-domain

⇒ f(x) is an onto function.

Now f(x) is an onto function but not one-one.

So f(x) is not a bijective function.

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Composition of two functions

Let f : A →B and  g : B →C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof : A →C given by

gof(x) = g[f(x)],  ∀ (for all)  x ∈ A

Composition of two functions exists only if co-domain of first function is equal to the domain of second function

In the above figure function f is defined on the element of set A (i.e.  x), function g is defined on the elements of set B (i.e f(x) ) so composition from set A to set C becomes gof or g[f(x)]

Bijective functions:

The functions which are one-one and onto both are called bijective functions.

Invertible functions and existence of inverse

If a function f is one-one and onto then it is an invertible function.

If a function is an invertible function then inverse of f exist and is denoted by f^-1.

If f is invertible the f must be one-one and onto, conversely if f is one-one and onto  then f must be invertible. If f is invertible then f^-1 exists.

Let f(x) = y is an invertible function then f^-1 exists and we can write f^-1(y) = x

Example : f(x) = { (a, 1), (b, 2), (c, 3), (d, 4)}, this can be written as

f(a) = 1, f(b) = 2,  f(c) = 3,  f(d) = 4

If f is invertible then we can write

f^-1(1) = a,  f^-1(2) = b,  f^-1(3) = c,  f^-1(4) = d

If f and g are two invertible functions then their composition gof is also invertible.

Give an example of a relation which is

i) Symmetric but neither reflexive nor transitive

Ans  R = { (1,2), (2,1)}

ii) Transitive but neither reflexive nor symmetric

Ans:  R = { (1,1), (2,2), (1,2), (2,1), (3,4)}

iii) Reflexive and symmetric but not transitive.

Ans: R = {(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)}

iv) Reflexive and transitive but not symmetric.

Ans: R = {(1,1), (2,2), (1,2), (2,1), (2,3), (3,3)}

v) Symmetric and transitive but not reflexive

{(5,6), (6,5),(5,5) }

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