HOW TO UNDERSTAND RELATIONS AND FUNCTIONS ,INVERSE OF A FUNCTION
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Today we are going to discuss Relations and Functions , "How to understand Relations and Functions, Inverse of a Function" under the topic Relations and Functions.
Today we are going to discuss Relations and Functions , "How to understand Relations and Functions, Inverse of a Function" under the topic Relations and Functions.
Ordered-Pair Numbers :-
Ordered-pair number is written within a set of parentheses and separated by a comma.
For example, (5, 6) is an ordered-pair number; the order is designated by the first element 5 and the second element 6. The pair (3, 6) is not the same as (6,3) because they have different order. Sets of ordered-pair numbers can represent relations or functions.
Example of ordered pair :
(3,8),(2,1),(7,6)
Relation
A relation is a set of ordered-pair numbers.
consider the following table
| Number of Students |
1 | 2 | 3 | 4 | 5 | 6 |
| Marks Obtained |
96 | 98 | 97 | 78 | 77 | 86 |
In the above table the numbers of students and marks obtained by them is a relation and can be written as a set of ordered-pair numbers.
A= {(1, 96), (2, 98), (3, 97), (4, 88),(5,77),(6,86)}
When we collect all the elements written in 1st column of the ordered pairs and placed in a set then the set so formed is called Domain of the relation.
The domain of A= {1, 2, 3, 4,5,6}
The domain of A= {1, 2, 3, 4,5,6}
As all the elements written in 2nd column of the ordered pairs and placed in a set then the set so formed is called Range of the relation.
The range of A = { 96, 98, 97,88 , 77, 86}
Function
A function is a relation in which every first element in ordered pairs have unique second element associated with them. Second elements may or may not be same.
we can better understand this concept with the help of this video
Example
{(1, 2), (2, 3), (3, 4), (4, 5),(5,6)} is an example of function
{ (1, 2), (2, 3), (3, 4), (4, 5),(5,6) } is a function because all the first elements are different.
Example
{(1, 3), (3, 3), (2, 1), (4, 2)} is an example of function
{(1, 3), (2, 3), (2, 1), (4, 2)} is a function because all the first elements are different.
Example
{ (1, 6), (2, 5), (1, 9), (4, 3) } is not an example of function
As in {(1, 6), (2, 5), (1, 9), (4, 3)} the element "1 " appeared twice .
Example
{(2, 15), (3, 15), (4, 15), (5, 13),(6,18)} is an example of function
As in {(2, 15), (3, 15), (4, 15), (5, 15)} all the first elements are different.
Example
{(1, 1), (-1, 1),(2,4),(-2,4), (3, 9), (-3, 9),(4,16),(-4,16)} is an example of function although the element "1" and "-1" ,"2" and "-2" , "3" and "-3" ,"4", "-4" have same images. This is an example of many one function.
Question:- Find x and y if:
(i) (5x + 3, y) = (4x + 5, 2)
(ii) (x – y, x + y) = (8, 12)
(iii) ( 2x-y , y+5 ) = ( -2,3 )
(iii) ( 2x-y , y+5 ) = ( -2,3 )
Solution
(iii) So By the equality of ordered pair elements
2x-y =-2 , y+5 = 3
2x = -2+y , y = 3-5
2x = -2+y , y = -2
Putting the value of y in 1st Equation ,we get
2x = -2 - 2
2x = -4
x = -2
so x= -2 and y =-2
A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.
1 ) Let B be the set of all triangles in a plane with R a relation in B given by
A function f : X → Y is defined to be one-one (or injection ), if the images of distinct elements of X under f are distinct, i.e., for every x, y ∈ X, f (x) = f (y) implies x = y. Otherwise, f is called many-one.
2 Function f : N → N, given by f (x) = 2x, is one-one but not onto.Because the elements have only one and unique image under f Therefore it is one one function .But not all elements of N have image under f
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
fof(x) = (16x + 12 + 18x -12 ) / ( 24x + 18 - 24x +16)
(1) Given (5x + 3 , y) = (4x + 5, 2)
So By the equality of ordered pair elements,
1st element of the ordered number written on the left hand side will be equal to the 1st element of the ordered pair number written on the right hand side . Therefore
So By the equality of ordered pair elements,
1st element of the ordered number written on the left hand side will be equal to the 1st element of the ordered pair number written on the right hand side . Therefore
5x + 3 = 4x + 5 and y = 2
5x-4x = 5 - 3 and y = 2
x = 2 and y = 2
5x-4x = 5 - 3 and y = 2
x = 2 and y = 2
(ii) So By the equality of ordered pair elements
x – y = 8 and x + y = 12
x – y = 8 and x + y = 12
Solving these two equations for x and y
2x =20 and 10+ y =12
x=10 y = 2
(iii) So By the equality of ordered pair elements
2x-y =-2 , y+5 = 3
2x = -2+y , y = 3-5
2x = -2+y , y = -2
Putting the value of y in 1st Equation ,we get
2x = -2 - 2
2x = -4
x = -2
so x= -2 and y =-2
Types of Relations
A relation R in a set A is called
(i) reflexive, if (a, a) ∈ R, for every a ∈ A,
(ii) symmetric, if (a, b) ∈ R implies that (a, b) ∈ R, for all a,b ∈ A.
(iii) transitive, if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R, for all a, b,c ∈ A.
If you want to Read HOW TO UNDERSTAND BINARY OPERATIONS , RELATIONS AND FUNCTIONS || COMMUTATIVE || ASSOCIATIVE click here
Equivalence Relation
A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.
1 ) Let B be the set of all triangles in a plane with R a relation in B given by
R = {(T1, T2) : T1 is congruent to T2}. Then R is an equivalence relation.
2 ) Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by
R = {(a, b) : both a and b are either odd or even}. Then R is an equivalence
one-one Function
A function f : X → Y is defined to be one-one (or injection ), if the images of distinct elements of X under f are distinct, i.e., for every x, y ∈ X, f (x) = f (y) implies x = y. Otherwise, f is called many-one.
Onto Function
A function f : X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an
element x in X such that f (x) = y.
Example
1 Function f : R → R, given by f (x) = 2x, is one-one and Onto As all the elements have only one and uniqe image under f.
e. g . 1,3,5,7... are not the image of any elements of N under f so it is not onto function
Example
The function f : N → N, given by f (1) = f (2) = 1 and f (x) = x – 1,
for every x > 2, is onto but not one-one.
Solution
Since f is Not one-one, as f (1) = f (2) = 1.
But f is Onto, as given any y ∈ N, y ≠ 1,
Choose x = y + 1 s.t.
f (y + 1) = y + 1 – 1
f (y + 1) = y.
Also for 1 ∈ N,
we are given f (1) = 1
Inverse of a Function
A function f : X → Y is defined to be invertible, if there exists a function g : Y → X such that gof = IX and fog = IY. The function g is called the inverse of f and is denoted by f –1
Example
Let S = {1, 2, 3}. Determine whether the functions f : S → S defined as below have inverses. Find f , if it exists.
(a) f = {(1, 1), (2, 2), (3, 3)}
(b) f = {(2, 2), (3, 1), (4, 1)}
(c) f = {(1, 5), (3, 4), (2, 1)}
Solution
(a) It is to proved that f is one-one and onto Hence f is invertible with the inverse f –1 of f given by f –1 = {(1, 1), (2, 2), (3, 3)} = f.
(b) Since f (3) = f (4) = 1, f is not one-one, so that f is not invertible.
(c) Here f is one-one and onto, so that f is invertible with
f –1 = {(5, 1), (4, 3), (1, 2)}.
Composition of 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)), ∀ x ∈ A
Example
fof(x) = (16x + 12 + 18x -12 ) / ( 24x + 18 - 24x +16)
fof(x) = (34 x ) / ( 34)
fof(x) = x = I(x)
Example
Let f : {2, 3, 4, 5} → {3, 4, 5, 9} and g : {3, 4, 5, 9} → {7, 11, 15} be functions defined as f (2) = 3, f (3) = 4, f (4) = f (5) = 5 and g(3) = g(4) = 7 and g(5) = g(9) = 11. Find gof = ?
Solution
We are given
gof (2) = g (f (2))
= g(3)
= 7
gof (3) = g(f (3)
= g(4)
= 7,
gof (4) = g(f (4))
= g(5)
= 11
and gof (5) = g(f (5))
= g (5)
= 11
So gof ={(2,7),(3,7),(4,11),(5,11)}
Example
conclusion
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