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 = nPm = 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 .
⇒ Rf = Odd natural number and Even natural numbers.
⇒ Rf = 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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Very nice and helpful thanks
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